Doctorate Dissertation

博士論文

Sterile neutrino oscillation studies with the T2K far detector
Super-Kamiokande(T2K)

実験におけるスーパーカミオカンデ遠地検出器を用いたステライルニュートリノ振動の研究

A Dissertation Submitted for Degree of Doctor of Philosophy
July 2018

平成30年７月博士（理学）申請

Department of Physics, Graduate School of Science,
The University of Tokyo東京大学大学院理学系研究科物理学専攻
Tsui Ka Ming

徐 嘉明

Abstract
The T2K (Tokai-to-Kamioka) experiment is a long-baseline accelerator neutrino experiment in Japan. The muon (anti-) neutrino beam produced at J-PARC is detectedby a set of near detectors at 280 m from the target to determine the neutrino flux andinteraction rates, and the Super-Kamiokande (SK) water-Cherenkov far detector 295
km apart to measure the oscillated neutrino spectra. The experiment uses an off-axis
configuration that the far detector is at an angle of 2.5o from the beam center. This
intercepts a narrow neutrino beam of peak energy 0.6 GeV which corresponds to thefirst minimum (maximum) of νµ disappearance (νe appearance) probability. T2K isthus optimized to measure the standard 3-flavor oscillation parameters: ∆m232 , θ23 ,θ13 and δCP . On the other hand, T2K also has the potential to explore exotic physics
like sterile neutrino.
The existence of sterile neutrinos is still an open question. They are singlet fermions which can contribute to weak interactions only through mixing with active
neutrinos and may explain some observations which appear anomalous in the standard three-neutrino scenario. In this thesis, we perform a joint oscillation analysis
using both charged current (CC) and neutral current (NC) SK samples to constrain
the parameters in the 3+1 sterile oscillation model: ∆m241 , θ24 and θ34 . With T2K
Run 1-8 dataset of 2.22914 × 1021 protons-on-target (POT), we exclude sin2 θ24 > 0.1

iiand sin2 θ34 > 0.5 at 90% limit for ∆m241 > 0.1eV2 .

iii

Preface
In a large experiment like T2K, it is impossible for a person to participate in every
single part of the experiment. Therefore, any physics result or discovery is credited
to the T2K collaboration as a whole. Indeed, my dissertation topic, sterile neutrino
study in T2K far detector, has gone through numerous collaboration discussions
and reviews before becoming official, and it is not possible without the efforts of all
past and present members. Chapter 3 describes the experimental facility and theanalysis software that are developed by this large collaboration. But for the sake ofa dissertation, I would like to highlight my personal contributions below.
In the past, T2K has only released results of short baseline sterile neutrino analysis. My study is therefore the first T2K long baseline sterile neutrino analysis, with
T2K data collected until 2017.
Before we really initiated the sterile analysis in the collaboration, I had done somepreliminary studies, without the use of T2K official fitter, to estimate the sensitive
parameter space, and determine the types of samples and variables to be used in
the fitting. These included modifying the oscillation probability formulas to see theeffects on different samples, correlating the reconstructed variables with neutrinoenergy, and rough χ2 fit with event numbers only. After we gained the approval fromanalysis working group, we moved towards to a strict study.

iv

In Chapter 4 where the oscillation samples are described, I have re-generated the
MC data of neutral current oscillation samples in order to obtain all the necessary
truth variables for oscillation analysis.
In Chapter 5 where the analysis method is outlined, I have joined the working
group which is responsible for the development of the analysis code (P-theta). Specific to my sterile study, I have modified the code to include the sterile oscillation
framework, and the additional neutral current (NC) samples. I have implemented
the systematic parameters which are dedicated to the NC samples, in which wehave spent lots of discussions and efforts to determine what parameters to use andwhat values to set. I have also changed the fitting strategy (from marginalization tominimization) in order to optimize the usage of computational resource.
In Chapters 6 and 7, I have performed the sensitivity studies and data fit on myown. I have also done lots of checks and alternative studies, which are shown inAppendix B to I.
The analysis result is now official and has been presented in ICHEP 2018. Ourlong baseline sterile oscillation analysis is a complement to the previously released
short baseline analysis in T2K[1], and we have successfully produced the world’s best
constraint on sin2 θ24 for ∆m241 < 3 × 10−3 eV2 .

v

Acknowledgement
I would like to express my gratitude to my supervisor Prof. Kimihiro Okumura, for
his guidance and advices on my studies, and providing me the invaluable opportunity
to participate in SK and T2K. He is so nice and accepted me as a foreign student in
the group, which led to my fantastic days in Japan.
Also, I wish to thank Prof. Takaaki Kajita, Prof. Yasuhiro Nishimura, Ryosuke
Akutsu and other people in RCCN, who makes me feel warm as a family. In particular
Ms Keiko Watanabe and Ms Chieko Mashima saved me a lot of troubles for my poor
Japanese.
I am grateful to meet Prof. Yasuhiro Nakajima, Prof. Jun Kameda and otherpeople in Kamioka who helped me with my studies in SK; Dr. Wing Ma, Prof.
Christophe Bronner, and members in the P-theta group who helped me getting startwith oscillation analysis in T2K; Prof. Yusuke Koshio, Dr. Alexander Izmaylov andcollaborators in T2K-exotics who helped me developing the sterile analysis. I blessthey will continue publishing exciting results in the future.
Finally, I want to express my gratefulness and love to my family, especially mywife, Chiu Tung Cheng, who supports my decision of PhD study, and even cameto Japan to live with me one year later (also starting a PhD). She makes my life
complete and colorful, and I wish to bring her a wonderful life too.

viContents

1 Introduction

1

2 Standard and sterile neutrino oscillation model

5

2.1

2.2

3-flavor neutrino mixing . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.1.1

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.1.2

Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.1.3

Matter effect . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.1.4

Current measured values of neutrino mixing parameters . . . .

12

Sterile neutrino model . . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.2.1

Theoretical interest . . . . . . . . . . . . . . . . . . . . . . . .

19

2.2.2

Experimental hints at eV scale . . . . . . . . . . . . . . . . . .

23

2.2.3

3+1 sterile neutrino model . . . . . . . . . . . . . . . . . . . .

24

3 The T2K experiment
3.1

29

Neutrino beamline at J-PARC . . . . . . . . . . . . . . . . . . . . . .

30

3.1.1

Proton beamline . . . . . . . . . . . . . . . . . . . . . . . . .

30

3.1.2

Neutrino beamline . . . . . . . . . . . . . . . . . . . . . . . .

31

3.1.3

Off-axis beam configuration . . . . . . . . . . . . . . . . . . .

34

3.1.4

Neutrino flux simulation . . . . . . . . . . . . . . . . . . . . .

35

vii

3.2

3.3

3.4

Near detector complex . . . . . . . . . . . . . . . . . . . . . . . . . .

37

3.2.1

INGRID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

3.2.2

ND280 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

3.2.3

ND280 software . . . . . . . . . . . . . . . . . . . . . . . . . .

46

3.2.4

BANFF fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

Far detector Super-Kamiokande . . . . . . . . . . . . . . . . . . . . .

48

3.3.1

Cherenkov radiation . . . . . . . . . . . . . . . . . . . . . . .

49

3.3.2

Detector design . . . . . . . . . . . . . . . . . . . . . . . . . .

50

3.3.3

Detector simulator SKDETSIM . . . . . . . . . . . . . . . . .

52

3.3.4

Detector calibration . . . . . . . . . . . . . . . . . . . . . . . .

52

3.3.5

T2K data taking . . . . . . . . . . . . . . . . . . . . . . . . .

58

3.3.6

Event reconstruction . . . . . . . . . . . . . . . . . . . . . . .

62

Neutrino event generator NEUT . . . . . . . . . . . . . . . . . . . . .

65

3.4.1

Quasi-elastic neutrino-nucleon scattering . . . . . . . . . . . .

65

3.4.2

Single meson production . . . . . . . . . . . . . . . . . . . . .

67

3.4.3

Deep inelastic scattering (DIS). . . . . . . . . . . . . . . . .

68

3.4.4

Coherent pion production . . . . . . . . . . . . . . . . . . . .

69

3.4.5

Multi-nucleon (2p2h) interaction. . . . . . . . . . . . . . . .

70

3.4.6

Final state and secondary interactions . . . . . . . . . . . . .

70

4 Oscillation samples and event selections at SK

72

4.1

Date set summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

4.2

Charged current oscillation samples . . . . . . . . . . . . . . . . . . .

75

4.2.1

FHC/RHC 1Rµ samples . . . . . . . . . . . . . . . . . . . . .

75

4.2.2

FHC/RHC 1Re samples . . . . . . . . . . . . . . . . . . . . .

82

viii

FHC νe CC1π + sample . . . . . . . . . . . . . . . . . . . . . .

86

Neutral current oscillation samples . . . . . . . . . . . . . . . . . . .

90

4.2.3
4.3

4.4

4.3.1

FHC/RHC 2Rπ 0 samples. . . . . . . . . . . . . . . . . . . .

91

4.3.2

FHC NCγ de-excitation sample . . . . . . . . . . . . . . . . .

96

Event summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5 Oscillation analysis framework

107

5.1

Event binning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.2

Monte Carlo Predictions . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.3

5.4

5.2.1

Neutrino propagation modes . . . . . . . . . . . . . . . . . . . 109

5.2.2

Neutrino interaction categories

5.2.3

True neutrino energy binning . . . . . . . . . . . . . . . . . . 110

5.2.4

Input preparation . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.2.5

Oscillation probability calculation . . . . . . . . . . . . . . . . 112

5.2.6

Event rate calculation . . . . . . . . . . . . . . . . . . . . . . 112

Systematic parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.3.1

Beam flux parameters . . . . . . . . . . . . . . . . . . . . . . 114

5.3.2

Neutrino interaction and cross-section parameters . . . . . . . 116

5.3.3

Super-K detector uncertainties . . . . . . . . . . . . . . . . . . 120

5.3.4

Effect of systematic parameters . . . . . . . . . . . . . . . . . 135

Fitting method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.4.1

Likelihood calculation . . . . . . . . . . . . . . . . . . . . . . 138

5.4.2

Confidence level building . . . . . . . . . . . . . . . . . . . . . 142

6 Sensitivity study
6.1. . . . . . . . . . . . . . . . . 110

143

3-flavor oscillation fit . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
ix

6.2

3+1 oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.2.1

sin2 θ24 -∆m241 plane . . . . . . . . . . . . . . . . . . . . . . . . 148

6.2.2

sin2 θ24 -sin2 θ34 plane . . . . . . . . . . . . . . . . . . . . . . . 150

6.3

Effect of statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.4

Effect of systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

7 Data fit results

155

7.1

Global best fit points . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

7.2

3-flavor oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

7.3

3+1 sterile oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

8 Summary and prospect

161

Appendix

165

A SK event reconstruction algorithms

165

A.1 apfit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
A.2 fiTQun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
B Event rate tables

175

C Sterile oscillation probability with matter effect

184

D Differences between NH and IH at ∆m241 ∼ ∆m231

186

E Effects of θ14 , δ14 and δ24 on the fit

189

F Setting Exclusion limits of sterile parameters with the Gaussian CLs
Method193x

G Fitting non-Asimov data sets198

H Valid ∆m241 parameter space

202

I

205

Event spectra at sterile exclusion limits

J List of abbreviations and symbols

207

Bibliography

211

xi

List of Figures
2.1

The transition probability P (νµ → νe ) with matter effect observed at
SK as a function of cosine of zenith angle and neutrino energy. The
zenith angle can be converted to neutrino propagation length throughthe Earth’s matter, assuming the Earth is a sphere. Top (bottom)
shows the normal (inverted) hierarchy case, and left (right) shows theneutrino (anti-neutrino) case (Figure taken from [2]). . . . . . . . . .

2.2

11

Ratio of the observed ν̄e spectrum to expected assuming no oscillation
for KamLAND, with the 3-flavor best fit oscillation curve superimposed. L0 = 180 km is the flux-weighted average reactor baseline (Figure
taken from [3]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3

13

Predicted solar neutrino spectra[4], overlaid with predicted MSW P (νe →νe ) (green is from solar best fit, blue is from solar + KamLAND bestfit). Shown in red is the 1σ band of P (νe → νe ) from combined dataof SK and SNO. Shown in points are the measurements of P (νe → νe )from the 7 Be (green), the pep (light green) and the 8 B flux (red) byBorexino[5], and the pp (blue) and CNO (gold) extracted from [6–9].(Figure taken from [10]). . . . . . . . . . . . . . . . . . . . . . . . .

xii

14

2.4

Allowed contours of ∆m221 vs sin2 θ12 from solar neutrino data (green),
KamLAND data (blue), and the combined result (red). The dasheddotted lines show the SK + SNO result. The filled regions give the 1,
2, 3 σ confidence level contours. For solar analysis, 4, 5 σ confidence
level contours are also shown (Figure taken from [10]). . . . . . . . .

2.5

15

The 90% and 68% confidence levels in the sin2 θ23 − ∆m232 planefrom T2K, SK[11], NOνA[12], MINOS[13] and IceCube[14], assuming∆m232 > 0 (Figure taken from [15]). . . . . . . . . . . . . . . . . . . .

2.6

Measured reactor ν̄e spectral distortion at Daya Bay (Figure takenfrom [16]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.7. . . . . . . . . . . . . . . . . . .

18

Schematic of a possible heavy sterile neutrino production at hadroncolliders followed by its subsequent decay, suggested in [17]. . . . . . .

2.9

17

Illustration of the two possible mass hierarchies, and the effect of δCPon the mixing matrix element Uαi .

2.8

16

22

The disappearance probability P (νµ → νµ ) as a function of neutrinoenergy at 295 km. The black line shows that standard 3-flavor result,while the red line shows the sterile one with ∆m241 = 0.1eV2 andsin2 θ24 = 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

2.10 The appearance probability P (νµ → νe ) as a function of neutrinoenergy at 295 km. The black line shows that standard 3-flavor result,while the red line shows the sterile one with ∆m241 = 0.1eV2 andsin2 θ24 = 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

27

2.11 The overall active neutrino survival probability 1 − P (νµ → νs ) asa function of neutrino energy at 295 km. The black line shows thatstandard 3-flavor result, while the red line shows the sterile one with
∆m241 = 0.1eV2 , sin2 θ24 = 0.1 and sin2 θ34 = 0.5. . . . . . . . . . . . .

28

3.1

A schematic diagram of the T2K experiment (Figure taken from [18]). 30

3.2

The J-PARC proton beamline (copyright J-PARC). . . . . . . . . . .

31

3.3

Overview of the T2K neutrino beamline (Figure taken from [18]). . .

32

3.4

The predicted νµ flux at SK for operation at different horn currents.
Top shows the flux histogram from 0 to 3 GeV, bottom shows theratios from 0 to 10 GeV (Figure taken from [19]). . . . . . . . . . . .

3.5

The T2K unoscillated neutrino flux prediction at SK for the neutrino(left) and antineutrino (right) modes (Figure taken from [20]). . . . .

3.6

33

34

The predicted neutrino flux at SK without oscillation at an off-axis
angle 0.0o (black), 2.0o (blue) and 2.5o respectively (red). The y-axis
units are different among three angles. On the top and bottom shows
P (νµ → νµ ) and P (νµ → νe ) at 295km (Figure taken from [19]). . . .

3.7

36

The near detector complex. The on-axis INGRID modules are locatedon the bottom two levels, while the off-axis ND280 detector is on theupper level (Figure taken from [18]). . . . . . . . . . . . . . . . . . .

38

3.8

INGRID on-axis detector (Figure taken from [18]). . . . . . . . . . .

39

3.9

An INGRID module. On the left shows the tracking planes and ironplates, on the right shows the veto planes (black) (Figure taken from[18]).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.10 The Proton module (Figure taken from [18]).

xiv. . . . . . . . . . . . .

40
41

3.11 An exploded view of the ND280 off-axis detector (Figure taken from[18]).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

3.12 A schematic of PØD. The beam is coming from the left to the right(Figure taken from [18]).

. . . . . . . . . . . . . . . . . . . . . . . .

43

3.13 Simplified drawing showing the main features of a TPC. The outerdimensions are roughly 2.3 m×2.4 m×1.0 m (Figure taken from [18]).

44

3.14 Distribution of the energy loss as a function of the momentum in the
TPCs for negatively charged particles produced in neutrino interactions, compared to the expected curves for muons, electrons, protons
and pions (Figure taken from [21]). . . . . . . . . . . . . . . . . . . .

45

3.15 Schematic of the package structure of the ND280 Software Suite. Only
the most representative packages are shown (Figure taken from [18]).

47

3.16 A schematic diagram of the SK detector (Figure taken from [22]). . .

50

3.17 Event displays of T2K events in SK for (a) a muon-like ring and (b)an electron-like ring. The cylindrical detector is being unrolled onto aplane. Each colored point represents the charge a PMT hit. The whiteline shows the reconstructed cone. On the top right corner shows the
OD hit map. The white crosses are the location of the reconstructedvertex, and the diamond is the location where a ray from the event
vertex would intersect the detector wall if it heads in the direction of
the beam (Figure taken from [18]). . . . . . . . . . . . . . . . . . . .

51

3.18 Typical fitted water coefficient functions used in SK-MC, which are
tuned with data taken in April 2009 (Figure taken from [23]). . . . .

54

3.19 Momentum distribution of decay electrons (MeV) for the data (black
points) and MC (red) (Figure taken from [24]).
xv. . . . . . . . . . . .

55

3.20 Invariant mass distributions of π 0 events for the data (black points)
and MC (red). The solid lines show the Gaussian fitting lines (Figure
taken from [24]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

3.21 Distributions of p(p.e.)/p(θ) of low energy stopping muons for the data
(black points) and MC (red), at three p(θ) bins: 200 - 280 MeV/c,
280 - 360 MeV/c and 360 - 440 MeV/c (Figure taken from [24]).

. .

57

3.22 Distributions of momentum over range of high energy stopping muons
for the data (black points) and MC (red) with various ranges. From
the top left to bottom right plots, the ranges are 5 - 10 m, 10 - 15 m,
15 - 20 m, 20 - 25 m, 25 - 30 m and 30 - 35 m respectively (Figure
taken from [24]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

3.23 Time variations of the momentum over range and momentum of decay
electrons. The red dotted lines are linear fit functions, and the green
lines are the value of MC (Figure taken from [24]).

. . . . . . . . . .

60

3.24 ∆T0 distribution of all FC, OD and LE events within ±500 µs aroundthe expected beam arrival time for T2K Run 1-7. The histograms arestacked in that order (Figure taken from [20]).

. . . . . . . . . . . .

61

3.25 Vertex, direction and momentum resolutions compared between apfit
and fiTQun for SK-IV fully contained atmospheric neutrino MC. The
resolution is defined as the 68 percentile of the respective distributions(Figures taken from [25]). . . . . . . . . . . . . . . . . . . . . . . . .

63

3.26 Event breakdowns for the 1-ring e-like (top), 1-ring µ-like (middle) and
NC1π 0 (bottom) samples of the SK-IV fully contained atmospheric
neutrino MC. fiTQun results are on the left and apfit results on theright (Figures taken from [25]). . . . . . . . . . . . . . . . . . . . . .
xvi

64

3.27 Feynman diagram of a neutrino CCQE interaction. . . . . . . . . . .

66

3.28 CC cross-section predicted by NEUT version 5.3.2 for (a) νµ and (b)ν̄µ . The T2K flux spectra are shown with arbitrary units (Figures
taken from [26]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1

66

Accumulated number of POT as a function of date for beam good
spills (blue), SK + beam good spills in neutrino mode (red), SK +beam good spills in antineutrino mode (orange), and the SK deadfraction of POT (green = (blue-red-orange)/blue) (Figure taken from[27]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

74

The cumulative number of observed FC events as a function of POT
in neutrino mode (FHC) and antineutrino mode (RHC) for T2K Run
1-8. The red solid lines refer to expectation in case of constant event
rates for all runs (Figures taken from [27]). . . . . . . . . . . . . . . .

4.3

74

Time variation of the SK attenuation length measured by cosmic ray
through-going muon data, Grat shaded regions denote the Run 1-8
periods (Figure taken from [27]).

4.4. . . . . . . . . . . . . . . . . . . .

75

Distributions of FHC (left) and RHC (right) 1Rµ event selections at
each stage for Run 1-8. Blue arrows denote the selection criteria. The
MC expectation is calculated based on 3-flavor oscillation frameworkwith the world best-fit oscillation parameters and BANFF fit corrections (Figures taken from [27]). . . . . . . . . . . . . . . . . . . . . . .

xvii

77

4.5

Distributions of FHC (left) and RHC (right) 1Rµ event selections at
each stage for Run 1-8. Middle graphs show the π + rejection cut in
2D, with events below the dotted yellow line selected as νµ candidates,and bottom graphs show the distance from the cut line (Figures takenfrom [27]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.6

78

Number of events passing each 1Rµ selection stage for T2K Runs
1-8. The MC expectation is calculated based on 3-flavor oscillation
framework with the world best-fit oscillation parameters and BANFF
fit corrections (Figures taken from [27]). . . . . . . . . . . . . . . . .

4.7

79

Reconstructed vertex distributions of FHC (left) and RHC (right)
1Rµ events for Run 1-8, after passing all selection cuts. Top graphs
show the wall distribution with towall > 250 cm applied, and bottom
graphs show the towall distribution with wall > 50 cm applied. The
blue arrows indicating the full FV cut (Figures taken from [27]). . . .

4.8

80

Reconstructed vertex distributions of FHC (left) and RHC (right) 1Rµevents for Run 1-8, after passing all selection cuts. Top graphs show
the square of the distance of the vertices from the central vertical axisof the SK tank, and bottom graphs show the vertical position of thevertices (Figures taken from [27]). . . . . . . . . . . . . . . . . . . . .

4.9

81

The cumulative number of observed 1Rµ events (blue) as a function of
POT in FHC and RHC for T2K Run 1-8. The red solid lines refer to
expectation in case of constant event rates for all runs (Figures takenfrom [27]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

4.10 Reconstructed neutrino energy distribution of 1Rµ samples for T2K
Runs 1-8 (Figures taken from [27]). . . . . . . . . . . . . . . . . . . .
xviii

82

4.11 Distributions of FHC (left) and RHC (right) 1Re event selections at
each stage for Run 1-8 (Figures taken from [27]). . . . . . . . . . . . .

84

4.12 Distributions of FHC (left) and RHC (right) 1Re event selections at
each stage for Run 1-8. Middle graphs show the π 0 rejection cut in
2D, with events below the dotted yellow line selected as νe candidates,and bottom graphs show the distance from the cut line (Figures takenfrom [27]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

4.13 Number of events passing each 1Re selection stage for T2K Runs 1-8(Figures taken from [27]). . . . . . . . . . . . . . . . . . . . . . . . .

86

4.14 Reconstructed vertex distributions of FHC (left) and RHC (right)
1Re events for Run 1-8, after passing all selection cuts. Top graphs
show the wall distribution with towall > 170 cm applied, and bottom
graphs show the towall distribution with wall > 80 cm applied. The
blue arrows indicating the full FV cut (Figures taken from [27]). . . .

87

4.15 Reconstructed vertex distributions of FHC (left) and RHC (right) 1Re
events for Run 1-8, after passing all selection cuts. Top graphs show
the square of the distance of the vertices from the central vertical axisof the SK tank, and bottom graphs show the vertical position of thevertices (Figures taken from [27]). . . . . . . . . . . . . . . . . . . . .

88

4.16 The cumulative number of observed 1Re events (blue) as a function of
POT in FHC and RHC for T2K Run 1-8. The red solid lines refer to
expectation in case of constant event rates for all runs (Figures takenfrom [27]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

4.17 Reconstructed neutrino energy distribution of 1Re samples for T2K
Runs 1-8 (Figures taken from [27]). . . . . . . . . . . . . . . . . . . .
xix

89

4.18 Feynman diagram of a CC1π + interaction . . . . . . . . . . . . . . .

89

4.19 Distributions of FHC νe CC1π + event selections at each stage for Run
1-8. Bottom left graph shows the π 0 rejection cut in 2D, with events
below the dotted yellow line selected as νe candidates, and bottom
right graph shows the distance from the cut line(Figures taken from[27]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

4.20 Number of events passing each FHC νe CC1π + selection stage for T2K
Runs 1-8 (Figures taken from [27]). . . . . . . . . . . . . . . . . . . .

91

4.21 Reconstructed vertex distributions of FHC νe CC1π + events for Run
1-8, after passing all selection cuts. Left graph shows the wall distribution with towall > 270 cm applied, and right graph shows the towall
distribution with wall > 50 cm applied. The blue arrows indicating
the full FV cut (Figures taken from [27]). . . . . . . . . . . . . . . . .

91

4.22 Reconstructed vertex distributions of FHC (left) and RHC (right) 1Re
events for Run 1-8, after passing all selection cuts. Top graphs show
the square of the distance of the vertices from the central vertical axisof the SK tank, and bottom graphs show the vertical position of thevertices (Figures taken from [27]). . . . . . . . . . . . . . . . . . . . .

92

4.23 Reconstructed neutrino energy distribution of FHC νe CC1π + sample
for Runs 1-8 (Figures taken from [27]). . . . . . . . . . . . . . . . . .

92

4.24 Number of ring distribution of all apfit FCFV multi-ring events forRuns 1-8 (Figures taken from [27]). . . . . . . . . . . . . . . . . . . .

93

4.25 Reconstructed π 0 invariant mass and momentum of 2Rπ 0 samples forRuns 1-8 (Figures taken from [27]). . . . . . . . . . . . . . . . . . . .

94

4.26 True neutrino energy distribution of 2Rπ 0 samples. . . . . . . . . . .

95

xx

4.27 Correlations between neutrino energy and π 0 invariant mass/momentum
for FHC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

4.28 The binding energy spectra of 16 O spectral function. The two peaks
correspibd to the 1p1/2 -hole state (ground state) and 1p3/2 -hole state
respectively. Neutron levels are deeper than protons’ by about 3.54
MeV (Figure taken from [28]).

. . . . . . . . . . . . . . . . . . . . .

state.
4.29 De-excitation modes of the 16 O proton hole (s1/2 )−1
p

97

Only

gamma rays with Eγ > 3 MeV are shown (Figure taken from [29]). . . 100
4.30 Distributions of G2V vs. G2A of NCγ de-excitation events with reconstructed energy between 4.5 and 5.0 MeV. The left plot shows the
distribution of T2K beam νµ simulated events, while the right plotshows that of the off-timing data. The dotted lines show a cut at
ovaQ≡ G2V − G2A = 0.2 (Figures taken from [28]). . . . . . . . . . . . 103
4.31 Cut threshold of dwall, effwall and ovaQ as a function of reconstructedenergy for T2K Run4. Red points show the optimized cut values ineach energy bin, while red dash lines show the fitting lines (Figures
taken from [28]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.32 Distributions of reconstructed energy of NCγ de-excitation events forT2K Run 4 before and after selection cuts (Figures taken from [28]). . 104
4.33 Distributions of FHC NCγ de-excitation sample for Runs 1-4. . . . . 105
4.34 Correlation between reconstructed gamma energy and true neutrinoenergy of FHC NCγ de-excitation sample. . . . . . . . . . . . . . . . 106

xxi

5.1

Error size and correlations for all the 133 systematic parameters of
the analysis. Parameters are 1-50: beam flux, 51-72: neutrino interaction and cross-section, 73-132: Super-K detector, 133: SK p-scaleparameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.2

The normalized wall distributions of stopping cosmic muons for MC
simulation (red) and data (black) (Figures taken from [30]). . . . . . 124

5.3

Decay electron tagging efficiency and fake rate as a function of muon
momentum (left) and towall (right) (Figures taken from [30]). . . . . 124

5.4

Muon mis-identification rate for stopping muon data and MC, as afunction of muon momentum (left) and towall (right) (Figures takenfrom [30]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.5Graphical representation of the detector region bin as a function ofwall and towall, zoomed into small wall and towall regions (Figures
taken from [31]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.6

Overall SK topological cut errors as a function of visible energy. The
total error in each bin is the sum in quadrature of the “shift error”(bin content) and “fit error” (error bar). Top left: νe CC1e, bottom
left: νe CC other, top right: νµ CC1µ, bottom right: νµ CC other(Figures taken from [31]). . . . . . . . . . . . . . . . . . . . . . . . . 127

5.7

Error size and correlations for the SK detector efficiency uncertainties.
Parameters are 1-12: FHC 1Re, 13-18: FHC 1Rµ, 19-30: RHC 1Re,
31-36: RHC 1Rµ, 37-48: FHC νe CC1π + , 49-52: FHC 2Rπ 0 , 53-56:
RHC 2Rπ 0 , 57-60: FHC NCγ de-excitation. . . . . . . . . . . . . . . 132

xxii

5.8

Error size and correlations for FSI+SI+PN uncertainties. Parametersare 1-12: FHC 1Re, 13-18: FHC 1Rµ, 19-30: RHC 1Re, 31-36: RHC
1Rµ, 37-48: FHC νe CC1π + , 49-52: FHC 2Rπ 0 , 53-56: RHC 2Rπ 0 ,57-60: FHC NCγ de-excitation. . . . . . . . . . . . . . . . . . . . . . 135

5.9

Number of observed events for the 1Rµ samples obtained over 50k
throws of systematic parameters. On the left shows the predicted
number of events in all bins. The dotted lines show the ±1σ region
around the mean value. On the right shows the error envelope ineach sample bin. Blue boxes correspond to the ±1σ region aroundthe mean value, black histograms correspond to the nominal values
for default value of the systematic parameters. . . . . . . . . . . . . . 137

5.10 Number of observed events for the 1Re and νe CC1π + samples obtained
over 50k throws of systematic parameters. . . . . . . . . . . . . . . . 138
5.11 Number of observed events for the NC samples obtained over 50k
throws of systematic parameters. . . . . . . . . . . . . . . . . . . . . 139
5.12 Fractional change of number of events when a systematic parameteris increased by 1σ from nominal value. Parameters are 1-50: beamflux, 51-72: neutrino interaction and cross-section, 73-132: Super-K
detector, 133: SK p-scale parameters. . . . . . . . . . . . . . . . . . . 140

xxiii

6.1

Predicted spectra for each sample. Distributions for 1Rµ samples area function of Erec (GeV) whereas distributions for 1Re samples andνe CC1π + sample are a function of pe (MeV) and θ (degree). Distributions for 2Rπ 0 samples are a function of π 0 momentum (MeV)
whereas distributions for NCγ de-excitation sample are a function ofreconstructed γ energy (MeV). . . . . . . . . . . . . . . . . . . . . . . 145

6.2

Asimov sensitivity 2D confidence level contours in ∆m232 vs. sin2 θ23 fornormal and inverted hierarchy. The black cross marks the parameter
truth values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.3

Asimov sensitivity 2D confidence level contours in δCP vs. sin2 θ13 fornormal and inverted hierarchy. The black cross marks the parameter
truth values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.4

Asimov sensitivity 1D ∆χ2 in δCP for normal and inverted hierarchy.

6.5

Asimov sensitivity 2D confidence level contours in ∆m241 vs. sin2 θ24

148

for normal and inverted hierarchy. On the right shows our sensitivity
overlaid on limits to date from other experiments[32–39] (Figure takenfrom [32]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.6

Our analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

6.7

Asimov sensitivity 2D confidence level contours in ∆m241 vs. sin2 θ24
assuming NH. The exclusion limit is mostly determined by the νµsamples at large ∆m241 , and supplemented by νe and NC samples at
small ∆m241 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

6.8

Asimov sensitivity 2D confidence level contours in cos2 θ24 sin2 θ34 vs.
sin2 θ24 . On the right shows the 90% and 99% upper limits from
IceCube[40] and SK[35]. . . . . . . . . . . . . . . . . . . . . . . . . . 151
xxiv

6.9

Asimov sensitivity 2D confidence level contours of NH assuming current Run 1-8 POT and future POT. . . . . . . . . . . . . . . . . . . . 152

6.10 Asimov sensitivity 2D confidence level contours of NH showing theindividual effect of different sources of systematic uncertainty. . . . . 153
6.11 Asimov sensitivity 2D confidence level contours of NH in sin2 θ24 -∆m241
plane, showing the systematic effects of Xsec and SK parameters. . . 154
7.1

Observed event spectra overlaid on the Asimov prediction. . . . . . . 156

7.2

Data fit 2D confidence level contours in ∆m232 vs. sin2 θ23 for normal
and inverted hierarchy. . . . . . . . . . . . . . . . . . . . . . . . . . . 158

7.3

Data fit 2D confidence level contours in δCP vs. sin2 θ13 for normal
and inverted hierarchy. . . . . . . . . . . . . . . . . . . . . . . . . . . 159

7.4

Data fit 1D ∆χ2 plots in δCP for normal and inverted hierarchy. . . . 159

7.5

Data fit 2D confidence level contours in ∆m241 vs. sin2 θ24 for normal
and inverted hierarchy. On the right shows our results overlaid on
limit to date from other experiments[32–39] (Figure taken from [32]). 160

7.6

Data fit 2D confidence level contours in cos2 θ24 sin2 θ34 vs. sin2 θ24 .
On the right shows the 90% and 99% upper limits from IceCube [40]and SK [35]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

A.1 Illustration of the Hough Transform (Figure taken from [41]).

. . . . 166

A.2 Illustration of Cherenkov photon emission from a muon. When muon
loss energy in a distance dx, photons are emitted in the region dx ·sin θ + r · dθ (Figure taken from [42]).

xxv. . . . . . . . . . . . . . . . . 167

A.3 Likelihood separation of single-ring electron (left) and muon (right)
events for SK-IV atmospheric neutrino MC. The magenta lines show
the cut criteria for electron-muon separation (Figure taken from [25]). 172
A.4 Likelihood separation of νµ charged-current quasi-elastic (left) andNCπ + (right) events for T2K neutrino MC. The black lines show thecut criteria for muon-π + separation (Figure taken from [25]).

. . . . 172

A.5 Likelihood separation of the CC single electron (left) and NC single π 0(right) events in the SK-IV atmospheric neutrino MC. The magentalines show the cut criteria for electron-π 0 separation (Figure takenfrom [25]).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

A.6 Schematic diagram of ring counting in the multi-ring fitter of fiTQun.
The fifth and sixth rings are assumed to be e-like only. This diagramshows the procedure assuming the first ring to be e-like, and the sameprocedure is repeated for the case of first ring being π + -like (Figure
taken from [24]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
C.1 Posc with and without matter effect. Unless specified, all oscillation
parameters are the same as in Table 6.2. . . . . . . . . . . . . . . . . 185
D.1 Posc with sin2 θ24 = 0.2, ∆m241 = ∆m231 = 2.509 × 10−3 eV2 for NH and
IH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
E.1 “Daya Bay constraint” as a function of ∆m241 . . . . . . . . . . . . . . 190
E.2 Asimov sensitivity 2D confidence level contours assuming NH showing
the possible effects of θ14 , δ14 and δ24 . . . . . . . . . . . . . . . . . . . 192

xxvi

F.1 Exclusion limits of the sterile parameters drawn by CLs and fixed ∆χ2methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
G.1 Best fit χ2 distributions of both 3+1 sterile and 3-flavor oscillation
models for 2500 non-Asimov data sets. . . . . . . . . . . . . . . . . . 199
G.2 Best fit standard oscillation parameters for 2500 non-Asimov data sets. 200
G.3 Best fit sin2 θ24 -sin2 θ34 distribution for 2500 non-Asimov data sets. . . 200
G.4 Exclusion limits in the ∆m241 -sin2 θ24 plane from the Asimov and nonAsimov data sets, assuming NH. . . . . . . . . . . . . . . . . . . . . . 201
G.5 Exclusion limits in the cos2 θ24 sin2 θ34 -sin2 θ24 plane from the Asimov
and non-Asimov data sets, assuming NH. . . . . . . . . . . . . . . . . 201
H.1 P (νµ → νµ ) at ND at 0.6 GeV, as a function of ∆m241 with sin2 θ24 = 0.1.203
H.2 Asimov sensitivity 2D confidence level contours in ∆m241 vs. sin2 θ24 ,assuming NH. The blue lines show the nominal sterile analysis, andthe red lines show a test study of using PSK (νµ → νµ )/PN D (νµ → νµ )and PSK (νe → νe )/PN D (νe → νe ) in event rate calculation. Thisillustrates that out sterile fit is valid below ∆m241 = 0.3eV2 where the
ND sees little sterile oscillation. . . . . . . . . . . . . . . . . . . . . . 204
I.1

Asimov event spectra (black lines) together event spectra at Points#1 (blue), #2 (red) and #3 (green), assuming NH, overlaid on data
points (black points). The νe spectra are projected on the electron
momentum pe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

xxvii

List of Tables
2.1

Best-fit values for the three-flavor neutrino-mixing parameters from a
global analysis [43]. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1

12

The run period, POT and horn current of each T2K run. Run 5- Run 7 included periods of beam operation in both neutrino andantineutrino mode. (Table taken from [27]). . . . . . . . . . . . . . .

4.2

The spectroscopic factors of each nucleon state in 16 O (Table takenfrom [28]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3

73

98

Summary of the energy levels and gamma ray emission branching ratios for the 1p3/2 hole states (Table taken from [28]).

. . . . . . . . .

99

4.4

Number of events observed in each CC and NC oscillation samples. . 106

5.1

Binning used for reconstructed energy of 1Rµ samples. . . . . . . . . 108

5.2

Binning used for reconstructed lepton momentum and angle of 1Re
samples and FHC νe CC1π sample.

. . . . . . . . . . . . . . . . . . 109

5.3

Binning used for reconstructed π 0 momentum of 2Rπ 0 samples. . . . 109

5.4

Binning used for reconstructed γ energy of FHC NCγ de-excitation
sample.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

xxviii

5.5

Definition of neutrino interaction categories in this analysis. NEUT interaction modes of anti-neutrinos are negative. NCQE events (NEUT
mode 51,52) are classified as NC other. . . . . . . . . . . . . . . . . . 110

5.6

Binning used for true neutrino energy Eνtrue . . . . . . . . . . . . . . . 111

5.7

Weights applied to CC coherent pion events. . . . . . . . . . . . . . . 112

5.8

Binning of beam flux systematic parameters for neutrino mode. . . . 115

5.9

Binning of beam flux systematic parameters for anti-neutrino mode. . 115

5.10 Summary of the neutrino interaction parameters using the results of
ND280 fit. ∗ Errors are not constrained by the near detector. ˆ NonBANFF errors added for the sterile analysis. # NCQE error insteadof NC other error applies for FHC NCγ de-excitation sample.

. . . . 117

5.11 Binning of SK detector systematic parameters for 1Re and νe CC1π
samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.12 Binning of SK detector systematic parameters for 1Rµ samples. . . . 121
5.13 Binning of SK detector systematic parameters for 2Rπ 0 samples. . . . 121
5.14 Binning of SK detector systematic parameters for NCγ de-excitation
sample. Beam unrel. refers to background events that are not due toneutrino interactions, e.g. PMT dark noises, radioactive decay, etc. . 121
5.15 Criteria for event categorization based on final state information. The
number of charged pions (Nπ± ) and protons (NP ) only includes particles produced with momentum above Cherenkov threshold set at 156.0
MeV/c and 1051.0 MeV/c respectively. . . . . . . . . . . . . . . . . . 122
5.16 Decay electron efficiency and estimated systematic uncertainties. . . . 123
5.17 List of fiTQun cut variables used in the atmospheric neutrino fit.

xxix

. . 126

5.18 Input systematic errors for the ToyMC of SK detector efficiency uncertainties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.19 Input systematic errors of topological cuts (ring-counting, PID, π 0rejection cuts ) for 1Re and νe CC1π + samples. . . . . . . . . . . . . . 128
5.20 Input systematic errors of topological cuts (ring-counting, PID ) for
1Rµ samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.21 Input systematic errors of topological cuts for 2Rπ 0 samples. . . . . . 129
5.22 The uncertainties of primary gamma production for NCQE signal events.130
5.23 The detector response uncertainties for NCγ de-excitation sample. . . 131
5.24 The summary of SK detector efficiency errors for NCγ de-excitation
sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.25 NEUT FSI “1-sigma” parameter sets from the fit to π ± -C data[44]. . 134
5.26 Percent errors on the predicted event rates of CC samples. . . . . . . 136
5.27 Percent errors on the predicted event rates of NC samples. . . . . . . 136
5.28 Fixed ∆χ2f values used to build confidence level intervals . . . . . . . 142
6.1

Run and POT configuration for the eight oscillation samples. . . . . . 143

6.2

Oscillation parameters used to generate ToyMC data set. . . . . . . . 144

6.3

Allowed values of the 3-flavor parameters in the oscillation fit. . . . . 144

7.1

Event rates of T2K Run 1-8 data and Asimov data set generated usingparameters in Table 6.2. . . . . . . . . . . . . . . . . . . . . . . . . . 155

7.2

Best fit oscillation parameter values for both 3+1 sterile model and 3flavor model. On the right shows the results from standard oscillation
analysis [45]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

B.1 Event rate table for FHC 1Rµ sample in 3-flavor oscillation analysis.
xxx

175

B.2 Event rate table for FHC 1Re sample in 3-flavor oscillation analysis. . 176
B.3 Event rate table for RHC 1Rµ sample in 3-flavor oscillation analysis.

176

B.4 Event rate table for RHC 1Re sample in 3-flavor oscillation analysis. . 177
B.5 Event rate table for FHC νe CC1π + sample in 3-flavor oscillation analysis.177
B.6 Event rate table for FHC 2Rπ 0 sample in 3-flavor oscillation analysis. 178
B.7 Event rate table for RHC 2Rπ 0 sample in 3-flavor oscillation analysis. 178
B.8 Event rate table for FHC NCγ de-excitation sample in 3-flavor oscillation analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
B.9 Event rate table for FHC 1Rµ sample in 3+1 sterile oscillation analysis.179
B.10 Event rate table for FHC 1Re sample in 3+1 sterile oscillation analysis.180
B.11 Event rate table for RHC 1Rµ sample in 3+1 sterile oscillation analysis.180
B.12 Event rate table for RHC 1Re sample in 3+1 sterile oscillation analysis.181
B.13 Event rate table for FHC νe CC1π + sample in 3+1 sterile oscillation
analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
B.14 Event rate table for FHC 2Rπ 0 sample in 3+1 sterile oscillation analysis.182
B.15 Event rate table for RHC 2Rπ 0 sample in 3+1 sterile oscillation analysis.182
B.16 Event rate table for FHC NCγ de-excitation sample in 3+1 sterile
oscillation analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
G.1 Event rates of four non-Asimov data sets. . . . . . . . . . . . . . . . . 199

xxxi

Chapter 1
Introduction
When Wolfgang Pauli first proposed the existence of neutrino in the 1930s to explain
the missing energy in radioactive beta decay experiments, he described it as a “desperate remedy” and wrote “I have done something very bad today by proposing aparticle that cannot be detected; it is something no theorist should ever do.” Indeed
this mysterious “ghost particle” was a puzzle (and still is now). After more than
20 years of technological advancement, the first neutrino, precisely the electron antineutrino ν̄e , was discovered in a reactor-neutrino experiment[46] through the inverse
beta decay reaction:
ν̄e + p → e+ + n.

(1.1)

Following the first neutrino observations, muon neutrino νµ , which couples tomuon instead of electron, was discovered in an accelerator neutrino experiment in
1962[47]. While the third charged lepton, tau, was discovered in 1975[48], it takes
another 25 years for the DONUT experiment[49] to directly detect tau neutrino in
2000.
Neutrino physics has been advancing very quickly since its discovery. One of1

2
the most interesting topics is the phenomenon of neutrino oscillation, where one
neutrino flavor can change its identity on propagation and become another flavor at
detection. Actually the concept of neutrino oscillation was first introduced in 1957 byPontecorvo[50, 51], although at that time he was proposing the oscillation between
neutrinos and anti-neutrinos ν  ν̄. Subsequently, the ideas of neutrino mixing
and flavor oscillations were presented in 1962[52, 53], and the theory of neutrino
oscillation was finally developed in 1976[54–57]. Such framework is analogue to quark
mixing, and allows the violation of charge-parity (CP) symmetry in the leptonic
sector, which might be the answer to the observed matter-antimatter asymmetry in
our universe.
Another implication of neutrino oscillation is that neutrino must be massive.
This violates the Standard Model of particle physics, so a good understanding of
this process can help the development of physics beyond Standard Model.
There are many different types of neutrino oscillation experiments with various
neutrino sources and detector designs. They cover a diverse range of neutrino energy
and measure different sets of oscillation parameters. Their results are mostly consistent with the standard 3-flavor oscillation model, in which the three active neutrino
flavors νe , νµ and ντ can be expanded as superpositions of three mass eigenstates ν1 ,
ν2 and ν3 . The corresponding mixing angles θij and mass splittings ∆m2ij = m2i − m2j
are measured with a high precision. For example, solar and long baseline reactor neutrino experiments determine the solar mass splittings ∆m2sol = ∆m221 ' 2.3×10−3 eV2and sin2 θ12 . Atmospheric and long baseline accelerator neutrino experiments quantity the atmospheric mass splittings ∆m2atm = |∆m231 | ≈ |∆m232 | ' 7.5 × 10−5 eV2and sin2 θ23 . Meanwhile short baseline reactor neutrino experiments give us measurements on ∆m231 and sin2 θ13 .

3

On the other hand, several oscillation experiments have indicated the possibilityof the existence of a new mass splittings ∆m2 ∼ O(eV2 ) which cannot be accommodated in the 3-flavor model. One possible way to explain this is to introduce
one or more additional massive neutrinos which do not participate in the standard
weak interactions. They are called sterile neutrinos in contrast to the three “active”
neutrino species.
Apart from solving the oscillation problem, sterile neutrinos are also interesting inmany aspects because they are present in many extensions of the Standard Model andhave important implications on astrophysics, cosmology, etc. In e.g. the Neutrino
Minimal Standard Model (νMSM)[58–60], there are three sterile neutrinos which canbe dark matter candidate and explain the baryon asymmetry in the Universe.
In this thesis, we perform a search of sterile neutrino in T2K. T2K is a long baseline acceleration neutrino experiment which is designed for precise determinationon ∆m232 and sin2 θ23 through muon neutrino disappearance measurement. Combining with simultaneous electron neutrino appearance measurement, recent T2K data
have also showed some preference on non-zero CP phases and normal neutrino massordering[20]. The high quality data set at T2K has offered us opportunity to study
exotic physics, e.g. the existence of sterile neutrinos. We make use of T2K’s data at
the far detector Super-Kamiokande (SK) to search for sterile neutrino, specificallyunder the 3+1 sterile model.
In Chapter 2, we give a brief overview of the standard 3-flavor oscillation model and introduce the 3+1 sterile model. A description of the T2K experiment ispresented in Chapter 3, and the oscillation samples collected at SK are describedin Chapter 4. We explain our analysis method in Chapter 5, and show the resultsof sensitivity study with Monte Carlo in Chapter 6. Finally the data fit results are4illustrated in Chapter 7, followed by a summary section in Chapter 8.

Chapter 2Standard and sterile neutrino
oscillation model
2.1

3-flavor neutrino mixing

2.1.1

Motivation

In the charged-current (CC) weak interaction which is mediated by the W boson, theW boson is always coupled to a charged lepton (e, µ, τ ) and its corresponding neutrino(νe , νµ , ντ ). A neutrino flavor eigenstate is therefore defined by the participating
charged lepton in the CC interaction. For example, reactor neutrinos are producedby the beta decay interaction which involves electron, and hence are all electron
anti-neutrino,n → p + e− + ν̄e .

5

(2.1)

2.1. 3-flavor neutrino mixing

6

On the other hand, the main decay mode of charged pion is to muon and muon
neutrino:
π + → µ + + νµ ,(2.2)
π − → µ− + ν̄µ .
Lepton number (+1 for charged lepton and neutrino, -1 for their antiparticles) isconserved in these interactions, so ν and ν̄ are produced correspondingly.
When a pure neutrino flavor is produced at a source, we expect the same flavorinteraction anytime/anywhere at detection. However, experiments show that it is
not the case. Such behavior was first observed in the Homestake experiment[6]in late 1960s, where only one-third of solar νe was detected when compared withtheoretical calculations. While there were various possible explanations such as theinaccuracies in the solar model and experimental systematics, one interesting theory
stands out: the depletion could be due to the flavor oscillation of νe into νµ andντ inside the Sun. At first, this theory received little support because it requires
a massive neutrino which contradicts the Standard Model. However, as more andmore experiments came out, there were stronger evidences on the solar νe deficit.
Finally the SNO experiment[61] proves that solar neutrino oscillation indeed exists
by measuring both the νe flux and the total solar neutrino (νe + νµ + ντ ) flux at thesame time. A deficit in νe flux was observed, while total solar neutrino flux agreed
with the standard solar model, which implied νe changing flavor into νµ and ντ .
Independent experiments with different neutrino sources and detector designs,
such as Super-Kamiokande[62] and KamLAND[63], also strongly support the theoryof neutrino oscillation. The oscillation theory is now well-established, and experiments have moved from discovery of oscillation to precision measurement of oscilla-2.1. 3-flavor neutrino mixing

7

tion parameters.

2.1.2

Formalism

The physics of neutrino mixing is similar to quark mixing, where the neutrino flavor
states να are not identical to the mass eigenstates νi and their transformations aregoverned by a unitary mixing matrix. In the 3-flavor oscillation framework, να and νiare related by a 3 × 3 matrix called the Pontecorvo-Maki-Nakagawa-Sakata (PMNS)
matrix U :
νi =

X

να =

X

Uαi |να i ,

α

(2.3)
∗
Uαi
|νi i .

i

The PMNS matrix is parametrized by three mixing angles θ12 , θ13 , θ23 , and a CPphase δCP which allows charge-parity violation in the leptonic sector. It matrix canbe decomposed into three rotation matrices depending on θij : Ue1 Ue2 Ue3 



U =
U
U
U
µ2
µ3 
 µ1


Uτ 1 Uτ 2 Uτ 3




−iδCP
0
0   c13
0 s13 e1  c12 s12 0





 −s

=
0
c
s
0
1
0
c
0
23
23    12 12 




0 −s23 c23−s13 eiδCP 0
c13
0
0 1


−iδCP
c12 c13s12 c13
s13 e





iδCP
iδCP
=
−s
c
−
c
s
s
e
c
c
−
s
s
s
e
s
c
12 23 1312 23
12 23 13
23 13  ,
 12 23s12 s23 − c12 c23 s13 eiδCP −c12 s23 − s12 c23 s13 eiδCP
c23 c13(2.4)

2.1. 3-flavor neutrino mixing

8

where cij = cos θij and sij = sin θij .
There is actually an additional diagonal matrix multiplied at the end of U :iα1 /2

e

 0


0

0
eiα2 /2
0

0

0
.

1

(2.5)

The Majorana CP-violating phases α1 and α2 only matter if neutrinos are Majorana
particles, i.e. neutrinos are their own antiparticle, which violate lepton number
conservation. These phases are mostly searched by neutrinoless double beta decay
experiments. Since they have no effect on the neutrino oscillation probability, they
are dropped out for simplicity.
Using U , a neutrino flavor state propagation can be written as|να (t)i =

X

∗
Uαi
|νi i ei(~pi ·~x−Et) .

(2.6)

i

In the ultra-relativistic limit E  mi ,q
pi = E 2 − m2i
(2.7)

m2
≈E− i,
2E
and neutrino travels at almost speed of light L ≈ t, we can simplify Eq. 2.6 as|να (L)i ≈

X
i

m2
iL

∗
Uαi
|νi i e−i 2E .

(2.8)

2.1. 3-flavor neutrino mixing

9

The probability of detecting νβ at distance L is therefore
P (να → νβ ) = | hνβ |να (L)i |2XX∗∗ −i(m2i −m2j )L/2E=
Uαi
Uβi Uαj Uβj
e
i

j


∆m2ij L= δαβ − 4
4E
i>j


X
∆m2ij L∗∗+2=(Uαi Uβi Uαj Uβj ) sin
.
2E
i>j
X

∗
∗
<(Uαi
Uβi Uαj Uβj) sin2(2.9)

For anti-neutrino, we replace U by U ∗ .
The oscillation behavior is mostly controlled by the sin2

∆m2ij L
terms. The oscil4E

lation amplitude is determined by the combination of sine and cosine terms of themixing angles θij . The oscillation phase is∆m2ij L, thus by changing the neutrino
4E

energy and detection baseline, one can map out the entire oscillation spectrum andmeasure ∆m2ij . And if ∆m2ij = 0, i.e. all neutrinos are massless, there would be no
oscillation.
While δCP is attached to θ13 , it does not enter the formula unless all θij are
non-zero. If δCP = 0 or π, there would be no difference between P (να → νβ ) andP (ν̄α → ν̄β ). Otherwise there would be an asymmetry between neutrino and antineutrino:∆P (να → νβ ) = P (να → νβ ) − P (ν̄α → ν̄β )X∆m2ij L∗∗=4=(Uαi Uβi Uαj Uβj ) sin
.
2E
i>j

(2.10)

2.1. 3-flavor neutrino mixing

2.1.3

10

Matter effect

When neutrinos propagate in matter, they interact with the materials (mostly electrons) through charged-current (CC) and neutral-current (NC) interactions [64–66].
All neutrino flavors undergo NC interaction with the same amplitude,να + e− → να + e− ,(2.11)

the corresponding Hamiltonian is diagonal and symmetric, which just introduces anoverall phase factors for all flavors and has no physical significance.
On the other hand, only νe would experience CC interaction:
νe + e− → e− + νe .

(2.12)

νµ and ντ cannot undergo similar CC interactions due to the lack of the corresponding
leptons in ordinary matter. This introduces an extra term A in the Hamiltonian ofpropagating neutrino:
 

2
m1



1 
U  0H=
2E  

0


0
m22
0





0
A 0 0 
 † 



0
 U +  0 0 0 ,



2
m3
0 0 0

(2.13)

with√A = 2 2GF Ne E,(2.14)

where GF is the Fermi constant and Ne is the number density of electrons. For
anti-neutrino, U is replaced by U ∗ and A is replaced by −A.
To calculate the oscillation probability, we need to diagonalize the above Hamil-2.1. 3-flavor neutrino mixing

11

Figure 2.1: The transition probability P (νµ → νe ) with matter effect observed at SK
as a function of cosine of zenith angle and neutrino energy. The zenith angle can beconverted to neutrino propagation length through the Earth’s matter, assuming the
Earth is a sphere. Top (bottom) shows the normal (inverted) hierarchy case, andleft (right) shows the neutrino (anti-neutrino) case (Figure taken from [2]).
0tonian to get a new set of mass eigenvalues mi2 and mixing matrix U 0 which areenergy-dependent. Both the oscillation amplitude and oscillation phase are changed. Since the potential term A differs by a negative sign between neutrino andanti-neutrino, this introduces a δCP -like effect and has to be taken account in δCPmeasurement. It can also be shown that the two mass hierarchies are affected in different ways. For example, Fig. 2.1 shows the transition probability P (νµ → νe ) withmatter effect for a neutrino traveling through Earth’s matter, for normal/inverted
hierarchy neutrino/anti-neutrino case.
Matter effect can change the oscillation pattern significantly for high energy neutrinos propagating a long distance, e.g. solar neutrinos and atmospheric neutrino
travelling across the Earth. Yet, for T2K’s beam energy and baseline, matter effectis not significant, and is ignored in the sterile neutrino oscillation probability calcu-

2.1. 3-flavor neutrino mixing
Parameter∆m221|∆m232 |sin2 θ12
sin2 θ23
sin2 θ13

12
Best-fit(7.53 ± 0.18) × 10−5 eV2(2.45 ± 0.05) × 10−3 eV20.307 ± 0.013
0.51 ± 0.04
0.0210 ± 0.0011

Table 2.1: Best-fit values for the three-flavor neutrino-mixing parameters from a
global analysis [43].
lation. In Appendix C, we confirm that the matter effect is not important in our
sterile analysis.

2.1.4

Current measured values of neutrino mixing parameters

In order to determine all neutrino mixing parameters, θij , ∆m2ij and δCP , experiments
have been done to measure P (να → νβ ) at different L/E regions. Table 2.1 shows
the best-fit values of all currently known parameters from a global analysis. Since|∆m232 |  ∆m221 , we have ∆m232 ≈ ∆m231 , and currently no single experiment is able
to measure these two different oscillation cycles at the same time.
θ12 and ∆m221
The world’s most accurate measurement of θ12 and ∆m221 comes from a combined
fit of KamLAND, which is a long baseline reactor neutrino experiment measuring ν̄eflux from more than 50 commercial nuclear reactors in Japan and Korea, and global
solar neutrino data[10].
For long baseline reactor experiments, the survival probability of ν̄e can be written

2.1. 3-flavor neutrino mixing

13

Figure 2.2: Ratio of the observed ν̄e spectrum to expected assuming no oscillation
for KamLAND, with the 3-flavor best fit oscillation curve superimposed. L0 = 180
km is the flux-weighted average reactor baseline (Figure taken from [3]).
as:P (ν̄e → ν̄e ) ≈ 1 − sin2 2θ12 sin2

∆m221 L.
4E

(2.15)

Fig. 2.2 shows the KamLAND’s best-fit P (ν̄e → ν̄e ) spectrum, with a flux-weighted
average baseline of L0 = 180 km.
Solar neutrino experiments measures the solar νe flux produced in different protonproton chain reaction. As νe propagates out of the Sun, it experiences matter effect,
but with matter density gradually decreases from the Sun’s core to surface. This
leads to the MSW effect[64, 66, 67], where the slow decrease of matter density resonantly enhances the neutrino mixing. Fig. 2.3 shows the expected MSW survival
probability P (νe → νe ). The basic idea is, low energy neutrinos suffer little matter
effect and travel like those in vacuum, while high energy neutrinos are strongly affected and made mostly ν2 in production. Whether the energy is low or high depends

2.1. 3-flavor neutrino mixing

14

Figure 2.3: Predicted solar neutrino spectra[4], overlaid with predicted MSW P (νe →νe ) (green is from solar best fit, blue is from solar + KamLAND best fit). Shownin red is the 1σ band of P (νe → νe ) from combined data of SK and SNO. Shownin points are the measurements of P (νe → νe ) from the 7 Be (green), the pep (lightgreen) and the 8 B flux (red) by Borexino[5], and the pp (blue) and CNO (gold)
extracted from [6–9].(Figure taken from [10]).
on the relative size the matter potential A in the Hamiltonian in Eq. 2.13. The solar
matter density in the center region is ∼ 105 kg/m3 , and ∆m221 ≈ 7.5 × 10−5 eV2 , so
the transition from “low” to “high” energy regime occurs in the MeV region.
In the low energy limit which is essentially vacuum oscillations,

P (νe → νe ) ≈ 1 − sin2 2θ12 sin2

where sin2

1∆m221 L≈ 1 − sin2 2θ12 ,
4E
2∆m221 Lis averaged out due to the extremely large size of the Sun.
4E

(2.16)In thehigh energy limit,
P (νe → νe ) ≈ | hνe |ν2 i |2 ≈ sin2 θ12 .

(2.17)

By measuring the transition of P (νe → νe ) between the low energy and high energy
ends, one can then map out sin2 θ12 and ∆m221 .
The KamLAND + solar fit results are shown in Fig. 2.4. There is a slight tension
between KamLAND and solar measurements on ∆m221 , but KamLAND one domina-

2.1. 3-flavor neutrino mixing

15

Figure 2.4: Allowed contours of ∆m221 vs sin2 θ12 from solar neutrino data (green),
KamLAND data (blue), and the combined result (red). The dashed dotted lines
show the SK + SNO result. The filled regions give the 1, 2, 3 σ confidence level
contours. For solar analysis, 4, 5 σ confidence level contours are also shown (Figure
taken from [10]).
tes in the fit because of its much smaller errors. Also the combined fit gives a muchbetter constraint on sin2 θ12 .
θ23 and ∆m232
The first observation of atmospheric neutrino oscillation was done by the Kamiokande experiment[68]. It measured the atmospheric neutrino flux due to cosmic ray
interaction in the atmosphere. A deficit in muon neutrino flux was observed, which
was one of the earliest evidence of the existence of neutrino oscillations. The atmospheric neutrino oscillation is mostly due to sin2 θ23 and ∆m232 , where the survival
probability of νµ and ν̄µ can be written as:
P (νµ → νµ ) = P (ν̄µ → ν̄µ ) ≈ 1 − sin2 2θ23 sin2∆m232 L.
4E

(2.18)

Following the discovery in Kamiokande, series of atmospheric and long baseline

2.1. 3-flavor neutrino mixing

16

Figure 2.5: The 90% and 68% confidence levels in the sin2 θ23 − ∆m232 plane from
T2K, SK[11], NOνA[12], MINOS[13] and IceCube[14], assuming ∆m232 > 0 (Figure
taken from [15]).
accelerator neutrino experiments have been done to constrain sin2 θ23 and ∆m232 .
Modern experiments like T2K and SK have measured them to a high precision, as
shown in Fig. 2.5. What remain unknown are whether there is maximal mixing ofsin2 θ23 = 0.5, and the sign of ∆m232 .
θ13
The last mixing angle, θ13 , is determined by short baseline reactor experiments suchas Daya Bay[16], RENO[69] and Double Chooz[70]. With baseline of ∼ 1 km, they
are able to detect the ν̄e disappearance due to ∆m231 oscillation:
P (ν̄e → ν̄e ) ≈ 1 − sin2 2θ13 sin2

∆m231 L.
4E

Fig. 2.6 shows, e.g. ν̄e survival probability measured at Daya Bay.

(2.19)

2.1. 3-flavor neutrino mixing

17

Figure 2.6: Measured reactor ν̄e spectral distortion at Daya Bay (Figure taken from[16]).
Long baseline accelerator neutrino experiments like T2K, NOνA[12], MINOS[13]are also capable of θ13 measurements by measuring the electron neutrino appearance
from a pure neutrino beam:

2

2Pνµ →νe ≈ sin 2θ13 sin θ23 sin

2
2 ∆m314E.

(2.20)The precision is, however, a bit worse due to lower statistics and the uncertainty ofsin2 θ23 .
Unknown parameters
While most oscillation parameters are well determined, we have little knowledge
about two important variables: neutrino mass hierarchy and δCP . Fig. 2.7 illustrates
the two possible mass hierarchies, and the effect of δCP on the mixing matrix element
Uαi .
The two possible mass hierarchies are normal hierarchy (m1 < m2 < m3 ) and2.1. 3-flavor neutrino mixing

18

Figure 2.7: Illustration of the two possible mass hierarchies, and the effect of δCP on
the mixing matrix element Uαi .
inverted hierarchy (m3 < m1 < m2 ). Since |∆m232 |  ∆m221 , normal hierarchy (NH)corresponds to one larger mass and two smaller masses, while inverted hierarchy (IH)
corresponds to one smaller mass and two larger masses. Such mass pattern can be
crucial in, e.g. examining theory beyond Standard Model (BSM) that leads to theexistence of neutrino mass, determining the absolute neutrino mass scale, resolving
the Majorana nature of neutrino, etc.
The value of δCP is, as named, indicating whether there is CP violation in the
leptonic sector, which may be used to explain the observed matter-antimatter asymmetry in our universe.
Although these two important parameters are currently unknown, the small but
non-zero value of θ13 has made it possible to measure them through oscillation experiments. For example, in accelerator experiments, the electron neutrino appearance
channel is sensitive to δCP , which however has some degeneracy with mass hierarchy
since matter effect would enhance (suppress) the appearance probability in case of
NH (IH). Measurements in both neutrino and antineutrino channels can help breakthis degeneracy, but current experiments have not reach definitive conclusions yet.

2.2. Sterile neutrino model

19

T2K[71], NOνA[72] and MINOS[73] all seem to have some preference on maximum
CP-violation of δCP ' ± π2 , although there are tensions between the published results.
Atmospheric neutrino data from SK[74] also showed preference of δCP ' − π2 and favored NH by > 90%. Next generation experiments like Hyper-K[75], T2HKK[76]and DUNE[77] would be capable to unfold the mysteries with their exceptional sensitivity.

2.2

Sterile neutrino model

2.2.1

Theoretical interest

In the Standard Model (SM), the three active neutrinos that participate in weakinteractions are all left-handed (spins antiparallel to momenta), and all anti-neutrinos
are right-handed. The absence of right-handed neutrinos (left-handed anti-neutrinos)
means neutrinos cannot obtain mass through Higgs mechanism and thus neutrinos
are massless. Yet, the phenomenon of neutrino oscillation implies neutrinos must
be massive. Also, from current experiment data, neutrino mass is at most of eVorder[43], which is > 105 times smaller than the lightest fermion electron. The originof neutrino mass is by no doubt the missing ingredient in the SM and involves new
physics.
To solve the neutrino mass problem, the SM must be extended. One of thesimplest solutions would be the introduction of right-handed neutrinos. These righthanded neutrinos, however, are singlet fermions, so they do not interact via weakinteractions and are “sterile”. With the right-handed sterile neutrinos, it is possible
to include a Dirac mass term or Majorana mass term or both to generate neutrinomass.

2.2. Sterile neutrino model

20

In the case of single neutrino, the Dirac mass term is of the formmD (n̄L nR + n̄R nL ),(2.21)

so it is actually the coupling between the left-chiral neutrino state nL and the rightchiral neutrino state nR . If only the Dirac mass term exists, the coupling has tobe extremely small (compared with other particles) in order to explain the small
neutrino mass.
The Majorana mass term is of the formCmL n̄C
L nL + mR n̄R nR ,(2.22)

where the superscript C donates charge conjugate. Unlike other fermions, neutrino
is electrically neutral, so the Majorana mass term does not violate charge conservation and is allowed. However, the nL coupling would require a Higgs triplet withhypercharge +2 that does not exist in the SM.
A general mass term with both Dirac and Majoranan masses may be constructed
as








 mL mD   nL 
 ,
n̄C
L n̄R 
mD mR
nC
R


(2.23)and the mass eigenvalues (m1 , m2 ) are1
m1,2 =2q(mL + mR ) ± (mL − mR )2 + 4m2D .

(2.24)

2.2. Sterile neutrino model

21

In the special case of mL = 0 and mR  mD , we havem1 ≈ −

m2D,
mR(2.25)

m2 ≈ mR .
Therefore we have “light” active neutrino and a “heavy” sterile neutrino. This is theso-called (type-I) seesaw mechanism[78–82] and can be generalized to any numberof active and sterile neutrinos by replacing mL/R/D with the corresponding mass
matrices.
Because of its beauty and simplicity, sterile neutrino is present in many extensionsof SM, and it is in principal possible to have any number of sterile neutrinos with any
masses. One famous model is the Neutrino Minimal Standard Model (νMSM)[58–60],
where there are three right-handed neutrinos with masses smaller than the electroweak scale. In this model, the lightest sterile neutrino is a dark matter candidate
with O(keV) mass, and the two heavier sterile neutrinos need to have roughly degenerate masses of O(MeV) or O(GeV). The decay of heavy Majorana neutrinos would
generate a violation of lepton number conservation in the universe, which could thenconverted to baryon number non-conservation (sphalerons[83]) and therefore explain
the matter-antimatter asymmetry observed nowadays.
There are various possible ways to search for sterile neutrinos over a wide rangeof mass scale. In collider experiments like LHC, heavy sterile neutrinos could be
produced and decay into e.g. W± l∓ pairs (see Fig. 2.8). The search for the oppositesign di-lepton signal in association with the hadronic decay of W± is one possible strategy[17]. Sterile neutrino dark matter may be searched directly by looking
for a narrow decay line in X-ray spectra of dark matter dominated astrophysical2.2. Sterile neutrino model

22

Figure 2.8: Schematic of a possible heavy sterile neutrino production at hadroncolliders followed by its subsequent decay, suggested in [17].
objects[84–87], or indirect constraints can be formed through observations of structure formation at different spatial scales[88, 89]. Neutrinoless double decay experiments might give some hints too because the decay rate is sensitive to mN |UeN |2 ,where N ≥ 4 donates the sterile neutrino index. Nevertheless, we are particularlyinterested in O(eV) sterile neutrinos because of their observable signatures in oscillation spectra, which could be used to resolve some observed anomalies in oscillation
experiments.
For eV scale sterile neutrinos to exist, the type-I seesaw models need to assume
very small (10−12 ) Yukawa couplings between the active and sterile neutrinos, andat least two sterile neutrinos are needed to explain the observed mass splittings ofactive neutrinos. There are therefore extensions to the usual seesaw models thatallow light sterile neutrinos to appear “more naturally”, e.g. the flavor symmetry
models[90–92] where sterile neutrinos of eV and keV masses can co-exist by the split
seesaw mechanism[93] or Froggatt-Nielsen mechanism[94], and the “minimal extended type-I seesaw” model that adds a fermino singlet in addition to three sterile
neutrinos[91, 95, 96]. Apart from the correct mass-splittings, these models also provide sizable active-sterile mixing that matches experimental observations. Our long
baseline sterile neutrino oscillation search is sensitive to those models that produce
light sterile neutrinos (< 1 eV) with a small but measurable mixing with active
neutrinos (sin2 θ . 0.1).

2.2. Sterile neutrino model

2.2.2

23

Experimental hints at eV scale

While sterile neutrinos could possibly explain neutrino masses, fill the gap in SM,
and account for dark matter and matter-antimatter asymmetry, there are still noconcrete evidence to prove their existence. Here we briefly describe the neutrino
oscillation experiments that suggest O(eV) masses sterile neutrinos, which was thestarting ground of sterile neutrino theory. These eV sterile neutrinos could affect theoscillation spectra through mixing with the active neutrinos.
The first and most well-known hint came from the accelerator experiment LSND[97,98], where there was excess of ν̄e events from a pure ν̄µ beam. Such excess cannot
be explained by the 3-flavor transition of ν̄µ → ν̄e , and suggests a new ∆m2 ∼ 1eV2
which could be due to sterile neutrinos. The same parameter space has been studied
by subsequent experiments like MiniBooNE[38] and MINOS[32]. Although most parameter space has been ruled out, it’s still possible that a small sterile mixing ofsin2 θ ∼ 10−3 exists.
There are also some interesting findings from reactor neutrino experiments[99].
Reactor experiments like Daya Bay have reported deficit in measured ν̄e flux withrespect to predictions. It is possible to explain this so-called reactor anomaly witheV sterile neutrino in addition to flux model inefficiency. Similarly, the GALLEXand SAGE Gallium solar neutrino experiments reported deficit in νe events duringcalibration processes with radioactive sources[100, 101].
While the Planck cosmological data is consistent with no sterile neutrino[102],
it does not completely rule out massive sterile neutrinos with small mixing angles.
Oscillation experiments are still promising way to search for eV sterile neutrinos.

2.2. Sterile neutrino model

2.2.3

24

3+1 sterile neutrino model

If eV scale sterile neutrinos indeed exist, we might be able to discover them throughneutrino oscillation experiments. As motivated by the results of LSND and subsequent sterile oscillation searches, we focus on the simplest 3+1 sterile neutrino
model with ∆m241 . 1eV2 . This is a good approximation of the general 3+N models with one eV scale sterile neutrino and multiple heavy sterile neutrinos (>keV) in
Sec. 2.2.1, where we can only see the spectral distortions due to additional m4 (unlessthe heavy neutrinos constitute another ∆m2 . 1eV2 which is very unlikely). In the
future we may extend our study to 3+2 or any 3+N model which has more oscillation
parameters and needs extra care on treating different combinations/scenarios.
In the 3+1 model, the flavor state να would be written as a superposition offour mass states νi , where ν4 is the sterile neutrino with mass m4 . The transformation requires a 4 × 4 unitary matrix similar to the PMNS matrix. We use the2.2. Sterile neutrino model

25

parameterization in [103] to decompose the matrix into constituent mixing angles:



U
U
U
U
e2
e3
e4
 e1



Uµ1 Uµ2 Uµ3 Uµ4 
U3+1 = 

U

U
U
U
 τ1
τ2
τ3
τ 4


Us1 Us2 Us3 Us4




−iδ14
1
0
0
0
1
0
0
0
c
0
0
s
e
14
14








−iδ24  
0 1


0
0  0c24
0 s24 e
0
1 0
0 



=



0 0 c
 0


s
0
1
0
0
0
1
0



34
34  

0 0 −s34 c34
0 −s24 eiδ24 0
c24−s14 eiδ14 0 0
c14




−iδ13
0 s13 e0  c12 s12 0 0
00 0  c13
1






0 c23 s23 0 01
00 −s12 c12 0 0



,


 0

0 −s
 −s eiδ13 0
c
0
0
1
0
c
0



  13
13
23
23




0
0
0
1
0
0 0 1
0
0
0 1
(2.26)
where δ13 is the usual δCP in the 3-flavor case. In addition to three new sterile mixing
angles θ14 , θ24 and θ34 , there are also two new CP phases δ14 and δ24 . In accelerator
experiments like T2K, we are mostly sensitive to θ24 and θ34 but not the sterile CP
phases. Also, from solar and reactor neutrino analysis [104], sin2 θ14 = |Ue4 |2 < 0.041
at 90% CL. We may conveniently set θ14 = δ14 = δ24 = 0 and simplify the sterile2.2. Sterile neutrino model

26

mixing matrix toc12 c13 −c c s − c c s s eiδ13
23 24 1212 24 13 23s12 (c34 s23 + c23 s24 s34 )U3+1 =  − c s (c c − s s s )eiδ13
12 13 23 34
23 24 34s12 (−s23 s34 + c23 c34 s24 )+ c12 s13 (c23 s34 + c34 s23 s24 )eiδ13

c13 s12s13 e

c12 c23 c24 − c24 s12 s13 s23 eiδ13− c12 (c34 s23 + c23 s24 s34 )− s12 s13 (c23 c34 − s23 s24 s34 )eiδ13c12 (s23 s34 − c23 c34 s24 )+ s12 s13 (c23 s34 + c34 s23 s24 )eiδ13−iδ13

The oscillation probability can be calculated by the same formula in Eq. 2.9, with
U replaced by U3+1 and raise the summation limit from i = 3 to i = 4.
In T2K, we are interested in 3 probability values: P (νµ → νµ ), P (νµ → νe ) and
1 − P (νµ → νs ), where νs may be regarded as a “sterile flavor state”.
The survival probability P (νµ → νµ ) strongly affects the muon neutrino disappearance samples. Eq. 2.28 shows P (νµ → νµ ) in an approximated form. With
non-zero θ24 , the overall ∆m231 oscillation amplitude would be reduced, and it’s inprinciple possible to probe the value of ∆m241 from the fast oscillation.
∆m231 L
∆m241 L− sin2 2θ24 sin24E
4E

(2.28)As an illustration, Fig. 2.9 shows P (νµ → νµ ) at the T2K-SK baseline L = 295 km
with and without sterile neutrino, where ∆m241 = 0.1eV2 and sin2 θ24 = 0.2.
The transition probability P (νµ → νe ) impacts the νe appearance samples. The
oscillation amplitude is reduced a bit with non-zero θ24 , as shown in Eq. 2.29 andFig. 2.10.
Pµe ≈ sin2 2θ13 cos2 θ24 sin2 θ23 sin2

∆m231 L
.
4E

0



c13 c24 s23
s24 




c13 (c23 c34 − s23 s24 s34 ) c24 s34 
.





−c13 (c23 s34 + c34 s23 s24 ) c24 c34 (2.27)

Pµµ ≈ 1 − sin2 2θ23 cos 2θ24 sin2(2.29)

Pµµ2.2. Sterile neutrino model

27

1

0.8

0.6

0.4

0.2

0
0

1

2

3

4

5

6

7

Neutrino Energy (GeV)

8

9

10

Pµe

Figure 2.9: The disappearance probability P (νµ → νµ ) as a function of neutrinoenergy at 295 km. The black line shows that standard 3-flavor result, while the redline shows the sterile one with ∆m241 = 0.1eV2 and sin2 θ24 = 0.2.

0.5

0.4

0.3

0.2

0.1

0
0

1

2

3

4

5

6

7

Neutrino Energy (GeV)

8

9

10

Figure 2.10: The appearance probability P (νµ → νe ) as a function of neutrino energy
at 295 km. The black line shows that standard 3-flavor result, while the red line shows
the sterile one with ∆m241 = 0.1eV2 and sin2 θ24 = 0.2.
The probability 1 − P (νµ → νs ) = P (νµ → νµ ) + P (νµ → νe ) + P (νµ → ντ )measures the total amount of all three active neutrinos that are not oscillated into
sterile neutrino. It is possible to measure this probability with the neutral current(NC) interaction samples as active neutrinos shall interact at the same rate. Any
deficit in such samples may be inferred as the sign of sterile neutrino oscillation. As
shown in Eq. 2.30, two oscillation modes can be seen (similar to P (νµ → νµ )). An

2.2. Sterile neutrino model

28

important reason to consider this probability is that it is a channel sensitive to θ34 .
1 − P (νµ → νs ) ≈ 1 − sin2 θ34 sin2 2θ23 sin2

∆m241 L∆m231 L− cos2 θ34 sin2 2θ24 sin2.
4E
4E
(2.30)

1-Pµs

Fig. 2.11 shows how the sterile neutrino might change the oscillation pattern.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0

1

2

3

4

5

6

7

Neutrino Energy (GeV)

8

9

10

Figure 2.11: The overall active neutrino survival probability 1 − P (νµ → νs ) as afunction of neutrino energy at 295 km. The black line shows that standard 3-flavor
result, while the red line shows the sterile one with ∆m241 = 0.1eV2 , sin2 θ24 = 0.1
and sin2 θ34 = 0.5.
In order to constrain θ24 and θ34 , it is preferable to measure multiple oscillation
probabilities at the same time. T2K provides a suitable environment for such measurement because of the rich sets of oscillation samples available at the far detector
SK.

Chapter 3
The T2K experiment
The T2K (Tokai-to-Kamioka) experiment is a long baseline accelerator neutrino oscillation experiment in Japan (Fig. 3.1). A high intensity muon neutrino beam is
produced by a 30 GeV proton beam hitting a graphite target, inside the J-PARC
facility in Tokai (Ibaraki Prefecture, east Japan). The neutrino beam is first measured by a set of near detectors at 280 m to determine the non-oscillated neutrino
flux and interaction cross-sections. Then the far detector Super-Kamiokande (SK)at Kamioka detects the oscillated neutrino events at 295 km away.
The neutrino beam is tuned in an off-axis configuration that its peak energy 0.6
GeV is at the first minimum of the νµ survival probability at 295 km. This allows
an accurate measurement of sin2 θ23 and ∆m232 with the νµ disappearance samples.
The νe appearance samples also give us some sensitivity on sin2 θ13 and δCP . There
are oscillation samples based on neutral current event selections as well, but they are
not used in standard oscillation analysis because of the lack of sensitivities on the
3-flavor oscillation parameters.
In the following, we describe in more details the T2K configuration, including29

3.1. Neutrino beamline at J-PARC

30

Super‐Kamiokande

Near Detectors

Mt. Noguchi‐Goro2,924 mJ‐PARC

Mt. Ikeno‐Yama
1,360 m

1,700 m below sea level

Neutrino Beam
295 kmFigure 3.1: A schematic diagram of the T2K experiment (Figure taken from [18]).
the neutrino beamline in Sec. 3.1, the two near detectors in Sec. 3.2, and the far
detector SK in Sec. 3.3. A brief description of the neutrino event generator, NEUT,
is provided in Sec. 3.4.

3.1

Neutrino beamline at J-PARC

3.1.1

Proton beamline

J-PARC (Japan Proton Accelerator Research Complex) is a multi-purpose facility
that makes use of high-intensity proton beam to generate various secondary particle
beams for a wide range of scientific purposes. The proton beam consists of threeparts: a linear accelerator (LINAC), a rapid-cycling synchrotron (RCS) and a main
ring (MR) (Fig. 3.2). The 330 m long LINAC accelerates negative hydrogen ionsH − to 181 MeV and injects into the RCS. Upon injection, the RCS converts H − toprotons by a charge-stripping foil, and accelerates them to 3 GeV at 25 Hz repetition

3.1. Neutrino beamline at J-PARC

31

Figure 3.2: The J-PARC proton beamline (copyright J-PARC).
rate and two bunches per cycle. About 5% of these bunches are further sent to the
MR for acceleration up to 30 GeV. For the neutrino beamline, a single extraction of
protons, known as a “spill”, happens at a frequency of 0.5 Hz. Each spill spans over
5 µs, which contains 8 proton bunches of length 100 ns separated from each otherby 560 ns. This narrow spill width helps reducing the detector background rate bya factor of 105 .

3.1.2

Neutrino beamline

The neutrino beamline (Fig. 3.3) is further divided into two sequential sections:primary and secondary beamlines. The primary beamline bends the proton beam
towards the direction of Kamioka and focuses it at the target by a set of normal
and superconducting magnets. Beam monitors are installed along the beamline to
examine the beam intensity, profile and loss during operation.

3.1. Neutrino beamline at J-PARC

32

Figure 3.3: Overview of the T2K neutrino beamline (Figure taken from [18]).
In the secondary beamline, protons hit the graphite target to produce mostly
secondary pions (and few kaons). Right after the targets, there are three magnetic
horns that guide the pions into a decay volume. The horns are powered by a 250
kA pulsed current which can be either positive or negative. The +250 kA “Forward
Horn Current” (FHC) is used to focus π + and produce mostly νµ beam:
π + → µ + + νµ .

The -250 kA “Reverse Horn Current” (RHC), on the other hand, focuses π − andproduces mostly ν̄µ beam:
π − → µ− + ν̄µ .
Fig. 3.4 illustrates that when the horns at operated at 250 kA, a maximum magnetic
field of 1.7 T is generated and increases the neutrino flux at SK by a factor of ∼ 17
at the peak energy.

3.1. Neutrino beamline at J-PARC

33

Figure 3.4: The predicted νµ flux at SK for operation at different horn currents. Topshows the flux histogram from 0 to 3 GeV, bottom shows the ratios from 0 to 10
GeV (Figure taken from [19]).
At the end of the decay volume, there is a beam dump that made up of thicklayers of graphite and iron. While neutrinos can go through it easily, hadrons andlow energy muons are stopped. Muons with momentum above ∼ 5.0 GeV, however,can pass through the beam dump and reach the muon monitor. The muon monitor isused to measure the muon distribution profile. Based on the knowledge of two-bodydecay into muon and neutrino, the neutrino beam intensity and direction can be
deduced from the muon kinematics on a bunch-by-bunch basis.
The intense proton beam inevitably produces some unwanted contaminations.
For example, in the FHC mode, interactions likeK + → π 0 + e + + νe

3.1. Neutrino beamline at J-PARC

34

Figure 3.5: The T2K unoscillated neutrino flux prediction at SK for the neutrino(left) and antineutrino (right) modes (Figure taken from [20]).
andµ+ → e+ + νe + ν̄µexist. The νµ /ν̄µ flux and νe /ν̄e contaminations are more precisely measured at thenear detectors. Fig. 3.5 shows the predicted neutrino and antineutrino flux at thefar detector.

3.1.3

Off-axis beam configuration

Interestingly, the neutrino beam is not pointing exactly towards the far detector (andone of the near detectors), but at an off-axis angle of 2.5o . This off-axis angle is used
to tune the peak energy and the width of the neutrino beam.
From relativistic kinematics, the neutrino energy Eν from a two-body π → µ + νµdecay is related to the pion momentum pπ , pion mass mπ and muon mass mµ by:Eν =m2π − m2µp,2( p2π + m2π − pπ cos θ)(3.1)

where θ is the neutrino emission angle relative the pion direction. Given a fixed pπ ,increasing θ reduces the outgoing Eν . Also, by differentiating Eq. 3.1 with respect3.1. Neutrino beamline at J-PARC

35

to pπ , there is an upper bound of Eν that can be obtained for non-zero θ, where:
pmax
=
π

mπ cos θ
,
sin θ

(3.2)

Eνmax =

m2π − m2µ.
2mπ sin θ(3.3)

The existence of upper bound thus narrows the Eν distribution with respect to zero
off-axis angle.
Fig. 3.6 shows the unoscillated neutrino flux predicted at SK, comparing thespectra at three different off-axis angles. Although the total neutrino flux is reduced,
the choice of the 2.5o off-axis angle shifts the neutrino energy peak to 0.6 GeV andmakes the spectrum much narrower which maximizes our sensitivity to measured
oscillation parameters.

3.1.4

Neutrino flux simulation

The neutrino flux simulation is done in three parts.
First, the primary interaction between the proton beam and the graphite target
is simulated by FLUKA[105], which agrees the best with external hadron production
data. Protons of 30 GeV are pointed to the target based on the measured proton
beam spatial distribution and divergence. All interactions and secondary particles
are tracked until they leave the target region.
Next, the particles are input into a GEANT3[106] simulation code called JNUBEAM, with hadronic interactions modeled by GCALOR[107, 108]. The particles
are propagated through the horns, where they are affected by the magnetic field andmay interact with the horn material, then through the decay volume until they inte-

3.1. Neutrino beamline at J-PARC

36

P(νµ→ νe)

P(νµ→ νµ)1
sin22θ23 = 1.0
sin22θ13 = 0.12∆m32 = 2.4 × 10-3 eV2

0.5

0.10

1

2

1

2

NH, δCP = 0
NH, δCP = π/2

IH, δCP = 0
IH, δCP = π/2

3

0.05

Φ295km
(A.U.)
νµ0
1

3

OA 0.0°OA 2.0°OA 2.5°

0.5

0
0

1

Eν (GeV)2

3

Figure 3.6: The predicted neutrino flux at SK without oscillation at an off-axis angle
0.0o (black), 2.0o (blue) and 2.5o respectively (red). The y-axis units are different
among three angles. On the top and bottom shows P (νµ → νµ ) and P (νµ → νe ) at
295km (Figure taken from [19]).

3.2. Near detector complex

37

ract or decay. When a neutrino is produced, it is forced to point to the direction of
SK or a random point in the near detector plane. The neutrino event information isstored, and the probability of pointing to that direction is saved as an event weight.
Finally, an event-by-event reweighting process is applied to all neutrino events,
which incorporates various hadron production datasets, including the NA61/SHINE
measurements on charged pion and kaon[109–111]. NA61/SHINE (SPS Heavy ion
and Neutrino physics Experiment) at the CERN SPS (Super Proton Synchrotron)
makes use of a 2 cm “thin target” and the 90 cm “T2K replica target” delivered by the
T2K Collaboration to measure the production cross-section of secondary particles.
The reweighting process properly corrects the production rates of secondary pions
and kaons at target materials, and their subsequent interaction rates which may
further produce tertiary hadrons or attenuate hadrons that decay to neutrino. The
neutrino flux and energy spectrum are obtained by summing the events with theassigned event weights.

3.2

Near detector complex

The neutrino beam is first measured by two near detectors at 280 m from the proton
target: INGRID and ND280. This baseline is short compared to the 3-flavor oscillation length, so we assume the measurements are on the unoscillated beam that canbe used to constrain the neutrino flux and interaction models. The two detectors are
situated in a pit of depth 37 m (Fig. 3.7).

3.2. Near detector complex

38

Figure 3.7: The near detector complex. The on-axis INGRID modules are locatedon the bottom two levels, while the off-axis ND280 detector is on the upper level(Figure taken from [18]).

3.2. Near detector complex

39

~10m
1.5m

Designed
Beam center

Y
X

Z

~10m
Figure 3.8: INGRID on-axis detector (Figure taken from [18]).

3.2.1

INGRIDINGRID (Interactive Neutrino GRID) is an array of iron/scintillator detectors that is
set on the center of the neutrino beam axis. The purpose of this on-axis detector is tomeasure precisely the neutrino beam direction and intensity with the large statistics
of interaction events in iron. The beam center can be measured with a precisionbetter than 10 cm, which correspond to an angular precision of 0.4 mrad.
As shown in Fig. 3.8, the INGRID detector is composed of 14 identical modules
arranged in a cross along the vertical and horizontal axes. The center of the cross,with two overlapping modules, sits at the center of the beam. There are also two
separate modules located at off-axis directions to measure the beam axial symmetry.

3.2. Near detector complex

40

Veto planes

1.24mElectronics boxes

Iron plates

Tracking planes

YXZ

Figure 3.9: An INGRID module. On the left shows the tracking planes and ironplates, on the right shows the veto planes (black) (Figure taken from [18]).
In an INGRID module (Fig. 3.9), there are nine iron plates and 11 tracking
scintillator planes arranged in a sandwich structure. Interactions outside the module
are rejected by the surrounding veto scintillator planes. The modules are calibrated
by cosmic ray, both on the ground surface and the ND280 pit underground.
In between the vertical and horizontal modules, there is an extra Proton Modulethat efficiently detects the muons and protons produced in neutrino interactions,especially in the quasi-elastic channel that can be compared with Monte Carlo simulations. It consists of scintillator planes but no iron plate so that most neutrino
interactions are on a carbon target (see Fig. 3.10). Also, it has finer grain scintillator
to improve tracking efficiency.

3.2.2

ND280

ND280 is the primary off-axis detector at the near detector complex. It is a magnetized tracking detector placed exactly on the SK direction. It serves three important
purposes. First, it measures the off-axis νµ flux for event rate prediction at SK.
Second, it determines the beam νe content which is a non-reducible background forνe appearance search. Third, it measures the νµ interaction cross-sections to reduce3.2. Near detector complex

41Power supply
Electronics box
Veto planes

1.2m

Y

Z

X

Shading panels

Tracking planes

Figure 3.10: The Proton module (Figure taken from [18]).
the errors due neutrino interaction models. To fulfill these purposes, the ND280 detector is designed carefully to reconstruct both the νµ and νe charged current (CC)and neutral current (NC) interaction events.
The detector is drawn schematically in Fig. 3.11. It consists of the PØD followedby the tracker TPC/FGD sandwich. They are surrounded by a metal frame container
called the “basket” and ECalan electromagnetic calorimeter (ECal). The outermost
part is a recycled UA1 magnet with scintillator which acts asa side muon range
detector (SMRD).

UA1 magnet
ND280 makes use of the old CERN UA1/NOMAD magnet to provide a dipole magnetic field of 0.2 T. This magnetic field bends the trajectory of charged particles andallows the TPC to determine their momenta and signs through measurement of track3.2. Near detector complex

42

Figure 3.11: An exploded view of the ND280 off-axis detector (Figure taken from[18]).
curvature. The magnet consists of water-cooled aluminum coils and a flux returnyoke. The field uncertainty is around 2 G at the nominal field of 0.2 T, helping
to reduce the systematic uncertainty of momentum determination down to 2% forcharged particles below 1 GeV/c.

PØD
The Pi-zero detector (PØD) is used to measure the rate of the neutral current interaction process with π 0 production,νµ + N → νµ + N + π 0 + X,

on a water (H2 O) target which is the same detector material at SK. The π 0 thendecays two gammas to give visible signals in the detector. Such NC single π 0 pro-

3.2. Near detector complex

43

Figure 3.12: A schematic of PØD. The beam is coming from the left to the right(Figure taken from [18]).
duction forms an important background in the νe appearance oscillation samples at
SK, when only one gamma is reconstructed in the event.
The schematic design of PØD is shown in Fig. 3.12. The central section is composed of planes of scintillator bars interleaved wifh fillable water target bags andbrass sheets. The water target can be filled or emptied, providing a subtraction
method to calculate the water cross-sections. The front (upstream ECal) and rear(central ECal) sections are alternating scintillator planes and lead sheets that improves containment of electromagnetic showers and creates a veto region to rejectparticles entering from interactions outside the PØD.

TPCs
The three time projection chambers (TPCs) are designed to have excellent resolution in charged particle track reconstruction, momentum determination and particle

3.2. Near detector complex

44

Figure 3.13: Simplified drawing showing the main features of a TPC. The outerdimensions are roughly 2.3 m×2.4 m×1.0 m (Figure taken from [18]).
identification. With these capabilities, they can measure event rate of νµ and νe indifferent interaction channels.
A simplified drawing of the TPC is shown in Fig. 3.13. Each TPC uses an argonbased mixture as drift gas, contained in an inner box. The surrounding outer box
contains CO2 as an insulating gas. A uniform electric field is formed by the copper
strip pattern on the inner box panels and the central cathode panel, roughly alignedwith the magnetic field provided by the UA1 magnet. When charged particles passthrough the TPCs, ionization electrons produced drift from the central cathode to oneof the readout planes. The readout planes are instrumented with bulk micromegas
detectors that multiply and sample the electrons. The pattern and arrival time of thesignals can be used to reconstruct the particle trajectory in the active drift volume.
Particle identification in TPCs is facilitated by measuring the energy loss per
distance by charged particles in the gas. From Monte Carlo simulation and test
beam studies, the relationships between energy loss and momentum for various charged particles are determined. Fig. 3.14 shows, e.g. such distribution for negatively
charged particles. With an energy resolution of about 7.8% for minimum ionizing3.2. Near detector complex

45

Figure 3.14: Distribution of the energy loss as a function of the momentum in the
TPCs for negatively charged particles produced in neutrino interactions, comparedto the expected curves for muons, electrons, protons and pions (Figure taken from[21]).
particles (MIPs), the TPCs can identify the particle identity with quite a high accuracy.

FGDs
The two fine grained detectors (FGDs) serve the purposes to provide target mass
for neutrino interactions and, at the same time, trajectory reconstruction of charged
particles from interaction vertex. The dimensions and weight of target material are
the same for the two FGDs: 2300 mm×2400 mm×365 mm and 1.1 tons. However,the first FGD is made up entirely of scintillators (total 5760) which act as both target
and tracker, while the second one only contains 2688 scintillator bars, with rest of
the space filled by 15 cm thickness of water. By comparing the interaction rates inthe two FGDs, cross-section measurements can be done on both carbon and watertargets.

3.2. Near detector complex

46

ECal
The ND280 ECal that surrounds the inner detectors (PØD, TPCs and FGDs) areelectromagnetic calorimeter aiming to reconstruct photons and other high energy
particles that exit the inner detector volume. It is made up of plastic scintillator
bars as active material and lead absorber sheets between layers to provide a nearhermetic coverage.
According to its usage, the 13 independent ECal modules are classified into 3
types. The six Barrel-ECal modules that surround the tracker volume and the downstream module (Ds-ECal) that covers the downstream exit of the tracker volume
are used to extract information for particle identification (electron-muon-pion separation). The six PØD-ECal modules that surround the PØD volume is for detecting
gammas that either do not convert in PØD active volume or produce showers only
partially contained with PØD. They also confirm the passage of charged tracks,
identify MIPs and veto incoming backgrounds that make reconstruction of π 0 easier.

SMRD
The SMRD consists of 440 scintillator modules being inserted into the air gaps inside
the UA1 magnet flux return yokes. It measures the momenta of muons at large
angles with respect to the beam, triggers cosmic ray muons that enter or penetrate
the detector, and identify events that occur in or around the magnet.

3.2.3

ND280 software

The ND280 software structure is shown schematically in Fig. 3.15. In the simulation part, there are interfaces with the neutrino beam simulation and the neutrino

3.2. Near detector complex

47

Figure 3.15: Schematic of the package structure of the ND280 Software Suite. Only
the most representative packages are shown (Figure taken from [18]).
interaction generation packages to generate events in the ND280 region based on theneutrino flux and the geometry and materials in the detector. A Geant4[112]-basedMonte Carlo software is used to simulate the particle interaction inside the detector
and also the detector response. For both data and MC, neutrino events are stored
in the format of “oaEvent” and pass through reconstruction processes under the framework of RecPack toolkit[113]. The full event information in the oaEvent files isfurther compiled into the ROOT[114] object “tree”, making it easier for end-users
to perform analysis.

3.2.4

BANFF fit

The BANFF fit is a fit to ND280 data to constrain the neutrino flux and crosssection uncertainties propagated to the oscillation analysis, developed by the Beam
And ND280 Flux extrapolation task Force (BANFF). The fit is done in two steps.

3.3. Far detector Super-Kamiokande

48

First, the priors for flux and cross-section model parameters are calculated basedon external data, and beam data from proton beam monitors and INGRID. Then,a maximum likelihood fit is performed on selected ND280 samples to constrain theflux and cross-section uncertainties.
The use of external data to extract priors of cross-section parameters is necessary because the ND280 measurements do not cover the full phase spaces of thecross-section models, and do not provide enough information on the kinematics
of secondary tracks, especially charged and neutral pions. MiniBooNE[115] andMINERνA[116] data are appropriate choices because they span similar neutrinoenergies on carbon-like targets.
A number of ND280 samples are chosen to measure the observed muon momentum
and cosine of muon angle. In neutrino beam mode, the νµ samples are divided intoCC0π, CC1π and CCOther. In anti-neutrino mode, there are ν̄µ CC 1-Track andCC N-Track samples, and also νµ CC 1-Track and CC N-Track samples to study theνµ contamination.
The parameterizations of the flux and cross-section uncertainties are described
detailedly in Sec. 5.3.

3.3

Far detector Super-Kamiokande

Super-Kamiokande (SK) is the world’s largest land-based, 50kton water Cherenkov
detector located deep under Mt. Ikenoyama, Gifu Prefecture, Japan. The mean rockoverburden is around 1000 m (2700 m water-equivalent) which reduces the cosmic
ray background by a factor of 105 . SK is 295 km west of the neutrino beam sourcein J-PARC and acts as the far detector in the T2K experiment.

3.3. Far detector Super-Kamiokande

49

Since its operation in 1996, SK has been running stably through four data-taking
periods SK-I to IV, and T2K takes place in SK-IV period. In addition to accelerator
produced-neutrinos, SK has also released numerous important results on measuring
the proton decay lifetime, atmospheric and solar neutrino oscillations, and dark
matter searches. Such rich scientific results can be credited to SK’s extremely large
detector size and satisfactory photo-coverage achieved with 13000 photomultipliertubes (PMTs).

3.3.1

Cherenkov radiation

When charged particle travels faster than speed of light in water (refractive index
nw = 1.34),
v≥

c
,
nw

(3.4)

it emits cone of Cherenkov photons which is light shock wave. Charged particles ofdifferent masses have their corresponding momentum threshold to emit Cherenkov
light. Electrons, muons and charged pions, which are of lighter masses, have smaller momentum thresholds of 0.57 MeV/c, 118 MeV/c and 156 MeV/c respectively.
Proton is much more massive and has a large momentum threshold of 1.07 GeV/c.
The Cherenkov opening angle θc is determined by the ratio of photon speed andparticle speed in water, i.e.
cos θc =

c
nw v

(3.5)

For highly relativistic particle, v ≈ c, we have the characteristic θc ≈ 42o in water.
The Cherenkov photons produce a ring-shaped hit pattern when they reach thedetector walls. By looking at the time and charge information of the PMT hits, one
can deduce variables like event vertex, particle momenta and directions.

3.3. Far detector Super-Kamiokande

50

Figure 3.16: A schematic diagram of the SK detector (Figure taken from [22]).

3.3.2

Detector design

The SK detector consists of the inner and outer detectors which are both cylindricalin shape and filled with pure water. Fig. 3.16 gives a schematic diagram of thedetector. The inner detector (ID) has a diameter of 33.8 m and a height of 36.2 m
which houses 11129 inward-facing 50 cm diameter PMTs. The outer detector (OD)
encloses the ID and is about 2 m thicker both radially and on the axis at both ends.
The OD contains 1885 outward-facing PMTs of 20 cm diameter mounted on the
inner walls. The ID and OD are separated by a 50 cm wide stainless steel scaffolded
covered by plastic sheets to ensure optical isolation between ID and OD. The ID
wall is lined with black plastic sheets to absorb photons that minimizes scattering.
The OD walls, on the other hand, are lined with a highly reflective material called
Tyvek R which reflects photons and increases their chance of entering OD PMTs.
The ID has a 40% PMT cathode surface coverage, and the PMTs have a combined
quantum and collection efficiency of about 20%. This enables the ID to reconstruct
a number of physical observables to infer the corresponding neutrino interactions.

3.3. Far detector Super-Kamiokande

51

Figure 3.17: Event displays of T2K events in SK for (a) a muon-like ring and (b) anelectron-like ring. The cylindrical detector is being unrolled onto a plane. Each colored point represents the charge a PMT hit. The white line shows the reconstructed
cone. On the top right corner shows the OD hit map. The white crosses are thelocation of the reconstructed vertex, and the diamond is the location where a ray
from the event vertex would intersect the detector wall if it heads in the direction of
the beam (Figure taken from [18]).
Also, the ID is capable of identifying the neutrino flavor in interactions. Whenνµ and νe undergo charged current quasi-elastic (CCQE) interaction, they produce
muons and electrons respectively. Muons have large mass that travel through thedetector almost straightly. This produces a clear, sharp ring pattern of PMT hits.
On the contrary, electrons with much lighter mass scatter more easily and produce
electromagnetic showers. The resulting Cherenkov light cones displace slightly withrespect to each other and produce a “fuzzy” ring pattern on the PMTs. Fig. 3.17
shows the event displays of a “muon-like” and a “electron-like” events due to T2K
neutrino beam.
The OD is mostly used for vetoing cosmic ray muons and other backgrounds,
and identifies high energy particles that escapes the ID. Also, the 2 m thick of water
acts as a shield to attenuate gammas and neutrons from the radioactive decays in3.3. Far detector Super-Kamiokande

52

surrounding rock. Due to its sparsely spaced PMTs, the OD cannot perform detailed
event reconstruction. Yet, events observed in OD can still be used for different
purposes, such as data-MC event rate comparison.

3.3.3

Detector simulator SKDETSIMSKDETSIM is the full detector simulator of SK that propagates particles across thedetector, simulates PMT responses and produces MC digitized PMT hit data. It isa GEANT3 based program in which most hadronic interactions are handled by the
GCALOR physics package. The propagation of Cherenkov light is taken by custom
package which considers absorption, Rayleigh scattering (symmetric) and Mie scattering (asymmetric, very forward angular distribution), with the model parameters
calibrated by laser measurements. The PMT photocathode surfaces are constructedin the detector geometry. When photon reach the PMT surfaces, the PMT responses are simulated with the measured quantum efficiency, gain and other electronic
properties. The overall performance of SKDETSIM is checked by comparing withcosmic ray samples, and discrepancy between MC and data is only of percent level.

3.3.4

Detector calibration

To properly describe the data collected in such a large detector, various calibration processes have to been done to understand the detector performance. Here we
describe three typical items: PMT, water and energy calibrations.

3.3. Far detector Super-Kamiokande

53

PMT
There are two gain calibrations for the PMTs: relative and absolute. The relative
gain is adjusted by changing the high voltage supply of each PMT so that they give
approximately the same responses to a Xe lamp light source. The absolute gain
calibration is to determine the conversion factor between the charges collected by
the PMTs to the number of incident photo-electrons (p.e.). The calibration source
contains nickel wires that capture neutrons emitted from a 252 Cf neutron source.
The process generates low energy gamma rays of 6 - 9 MeV. When placed in thecalibration portholes, the source only produces 50 - 80 PMT hits in the whole ID,
so each PMT hit is most probably due to one photon. This gives the single photoelectron charge distribution to calibrate the PMT absolute pain.
Another PMT relevant calibration is the relative timing calibration. Since each
PMT is connected to the DAQ system with slightly different lengths of cable, thesignal time is offset by various amounts. Also, it takes shorter time for strong signals
to exceed the detection threshold than the weak signals, this introduces a second
timing offset. The calibration uses a diffuser ball to emit uniform light in the centerof the tank, and records the time-charge (photon-electron) distributions for all PMTs,
which provides information for timing correction.
Finally, daily monitoring of the PMT performance can be done with cosmic ray
muon samples. Vertically downward going muons transversing the full length of the
ID are selected. Their energy depositions in water are roughly constant of around
2 MeV/cm. The observed charges in the PMTs, after taking into account the lightattenuation in water and the PMT acceptance, are fit to measure the water transparency and tune the PMT charge scale (gain).

3.3. Far detector Super-Kamiokande

54

Figure 3.18: Typical fitted water coefficient functions used in SK-MC, which are
tuned with data taken in April 2009 (Figure taken from [23]).
Water parameters
There are three parameters describing the light attenuation in water: photon absorption, symmetric scattering due to Rayleigh scattering, and asymmetric scattering due
to Mie scattering. The light attenuation length L is a function of wavelength λ which
can be described as:

L(λ) =

1αabs (λ) + αsym (λ) + αasy (λ)

,

(3.6)

where αabs , αsym , αasy are the absorption, symmetric scattering and asymmetric scattering coefficients respectively. While the light attenuation length is also measured
by the cosmic muon data, more accurate measurements are performed with a laser source which can give both the wavelength dependency and contributions fromeach of the three processes. Fig. 3.18 shows a typical result of the measurement ofabsorption and scattering coefficients.

3.3. Far detector Super-Kamiokande

55

Figure 3.19: Momentum distribution of decay electrons (MeV) for the data (black
points) and MC (red) (Figure taken from [24]).
Energy calibration
The momenta (and thus energy) of the detected, charged particles are reconstructedfrom the PMT collected charges. To better understand the accuracy and uncertaintyof the absolute energy scale, different sources of particles with a wide range of energy
are used in the calibration study.
The lowest energy source comes from decay electrons (∼ 50 MeV/c). Decay
electrons are searched in a time window of 1.5 µs to 8.0 µs after a stopping muon
event. Low energy (∼ 6 MeV) gamma rays from µ− capture on a nucleon are rejectedby imposing a cut on the number of PMT hits. There are also higher momentum
events (∼ 70 MeV/c) which are due to µ− capture on the 1st shell (K-shell) of an
oxygen atom. Fig. 3.19 shows the momentum distribution of the decay electrons, inwhich a good agreement is obtained between data and MC.
The second source is from NC single π 0 events, where the decay π 0 → γγ gives
two electron-like rings in the detector. The invariant mass of the π 0 events are3.3. Far detector Super-Kamiokande

56

Figure 3.20: Invariant mass distributions of π 0 events for the data (black points) andMC (red). The solid lines show the Gaussian fitting lines (Figure taken from [24]).
reconstructed by the formula:qminv = 2pγ1 pγ2 (1 − cos θ),(3.7)

where pγ1 and pγ2 are the momenta of the two gamma rays, and θ is the angle between
them. The mass distribution is compared with MC prediction (see Fig. 3.20). The
difference in Gaussian fitted peak positions estimates the error as (0.75 ± 0.25)%.
The third source is the low energy stopping muons in the detector (< 500 MeV/c).
For these low energy muons, momentum can also be reconstructed from the Cherenkov opening angle θc :m.
p(θ) = pn2w cos2 θ − 1(3.8)

Fig. 3.21 shows the ratios of the nominal reconstructed momentum p(p.e.) to p(θ) atseveral p(θ) bins, both for data and MC. The maximum error is (−1.905 ± 0.402)%.

The final, highest energy source is the high energy stopping muons (around 1 to

3.3. Far detector Super-Kamiokande

57

Figure 3.21: Distributions of p(p.e.)/p(θ) of low energy stopping muons for the data
(black points) and MC (red), at three p(θ) bins: 200 - 280 MeV/c, 280 - 360 MeV/c
and 360 - 440 MeV/c (Figure taken from [24]).

3.3. Far detector Super-Kamiokande

58

10 GeV/c). Downward muon tracks with track length greater than 5 m are selected,
in which the track length is approximately proportional to the muon momentum.
Fig. 3.22 shows the momentum over range distributions of the data and MC, between
the range of 5 m and 35 m in 5 m bins. A momentum loss of ∼ 2.2 MeV/c/cm
is observed in every bin. The errors are evaluated by comparing the fitted peak
positions assuming Gaussian distributions, and are all less than 1%.
The overall absolute energy scale error is defined as the largest error of all theerrors described above, which is 1.90%. Meanwhile, time variations of the energy
scale are monitored by the decay electrons and momentum over range measurements(see Fig. 3.23). While the momentum over range is almost stable, the decay electron
momentum is increasing yearly by (0.21 ± 0.03)%. A time variation error is assigned
as the ratio of the RMS over mean and is 0.88%. The final energy scale error isp(1.90%)2 + (0.88%)2 = 2.1%.

3.3.5

T2K data taking

To record the T2K induced neutrino events, the SK’s DAQ system has a trigger
in time with the beam spills produced at J-PARC. Each beam spill has a GPS
timestamp that is accessible by SK, where the DAQ system defines a software trigger
to record all the PMT hit information in a 1 ms time window around the beam arrival
time. The spill events are then stored for offline analysis later. Fig. 3.24 shows event
timing (∆T0 ) distribution for all categories of events.
An event is triggered if the ID detects ≥ 47 PMT his in a 200 ns time window,
which roughly corresponds to a 5 MeV electron, or the OD detector has ≥ 22 PMT
hits in a 200 ns time window. The event includes all the PMT hits within −5 ∼ +35µs
from the detected trigger timing.

3.3. Far detector Super-Kamiokande

59

Figure 3.22: Distributions of momentum over range of high energy stopping muons
for the data (black points) and MC (red) with various ranges. From the top left tobottom right plots, the ranges are 5 - 10 m, 10 - 15 m, 15 - 20 m, 20 - 25 m, 25 - 30
m and 30 - 35 m respectively (Figure taken from [24]).

3.3. Far detector Super-Kamiokande

60

Figure 3.23: Time variations of the momentum over range and momentum of decay
electrons. The red dotted lines are linear fit functions, and the green lines are thevalue of MC (Figure taken from [24]).

3.3. Far detector Super-Kamiokande

61

Figure 3.24: ∆T0 distribution of all FC, OD and LE events within ±500 µs aroundthe expected beam arrival time for T2K Run 1-7. The histograms are stacked inthat order (Figure taken from [20]).

3.3. Far detector Super-Kamiokande

62

The T2K events are categorized into three mutually exclusive samples. If OD
activities are identified, i.e. number of PMT hits is greater than 15 in the largest
OD hit cluster, it is classified as an OD event. Then, low energy (LE) events are
extracted if the total charge of ID PMT hits in 300 ns time window is less than 200
p.e., which roughly corresponds to a 20 MeV electron. An event is also classified as
LE if a single ID PMT hit constitutes more than half of the total p.e. observed. Such
events are usually due to radioactivity near the PMT. Finally, all remaining eventsare classified asfully contained (FC) events which have Cherenkov light deposited
extensively in the ID.

3.3.6

Event reconstruction

The conventional event reconstruction algorithm in SK is called apfit, which wasdeveloped back in the late 1990’s at the beginning of SK operation. The physicalproperties of a event are reconstructed in a step-by-step process that uses only thetime and charge information from hit PMTs.
Recently T2K has developed a new fitter called fiTQun, which employs a maximum likelihood method and uses information from both hit and non-hit PMTs. It
has better performance in reconstructing multi-ring events, and dedicated fitters forπ 0 and π + respectively. Now T2K oscillation samples are gradually switching from
apfit-based to fiTQun based reduction. Details of the two reconstruction algorithms
can be found in Appendix A.

Comparison between apfit and fiTQun perforamce
Fig. 3.25 shows the vertex, direction and momentum resolutions compared between
apfit and fiTQun, based on SK-IV fully contained atmospheric neutrino MC. Clearly3.3. Far detector Super-Kamiokande

63(a) Single-ring electron (left) and muon(right) vertex resolution.

(b) Single-ring electron (left) and muon(right) direction resolution.

(c) Single-ring electron (left) and muon(right) momentum resolution as a function oftrue momentum.

Figure 3.25: Vertex, direction and momentum resolutions compared between apfit
and fiTQun for SK-IV fully contained atmospheric neutrino MC. The resolution isdefined as the 68 percentile of the respective distributions (Figures taken from [25]).
fiTQun is better in every perspective.
Fig. 3.26 shows the event breakdowns of the 1-ring e-like, 1-ring µ-like and NC1π 0events by fiTQun and apfit. While the single ring event reconstruction efficiencies
are about the same for both fitters, it is obvious that fiTQun performs much better on identifying NC1π 0 events. In generally fiTQun has better performance in3.3. Far detector Super-Kamiokande

64(a) 1-ring e-like events.

(b) 1-ring µ-like events.

(c) NC1π 0 events.

Figure 3.26: Event breakdowns for the 1-ring e-like (top), 1-ring µ-like (middle) and
NC1π 0 (bottom) samples of the SK-IV fully contained atmospheric neutrino MC.
fiTQun results are on the left and apfit results on the right (Figures taken from [25]).
reconstructing multi-ring events. Also the π + fitter makes it possible to reconstruct
CC1π + events explicitly. Currently T2K has already used the fiTQun 1-ring e-likeand µ-like samples as the official oscillation analysis samples, and more fiTQun based
samples are under development.

3.4. Neutrino event generator NEUT

3.4

65

Neutrino event generator NEUT

NEUT[117] is a neutrino interaction simulation library that generates primary neutrinonucleon interactions and handles subsequent secondary interactions within the interacting nuclei. Its results are used as an input for both the ND280 and SK simulation codes to simulate the detector responses. This library was originally developed
for the Kamiokande experiment but has been continuously updated to support theSuper-Kamiokande and T2K experiments.
In NEUT, the following primary neutrino interactions are simulated: (quasi) elastic scattering, single meson production, coherent pion production, and deep
inelastic scattering. Recently a multi-nucleon neutrino interaction model has been
added to NEUT as well. The secondary interactions between the nucleus and any
products of the primary interactions, including both mesons and recoiling nucleons,are modeled by a particle cascade routine.

3.4.1

Quasi-elastic neutrino-nucleon scattering

Among all the primary interactions, the charged current quasi-elastic (CCQE) interactions constitute the majority of the events in the SK oscillation samples, wherea neutrino scatters off from an entire nucleon and resembles an elastic scattering
kinematics (see Fig. 3.27). Fig. 3.28 shows the total CC cross-sections predicted by
NEUT, where we can see CCQE interaction dominates at the beam flux peak.
The CCQE cross-sections on free protons (or hydrogen atoms in water) are basedon the Llewellyn Smith Model[118]. For CCQE interactions on bound nucleons in
16

O, the NEUT default is therelativistic Fermi gas model (RFG) in [119], where themomentum distribution of the nucleons follows the Fermi gas model, and the out-

3.4. Neutrino event generator NEUT

66

νl

l

−

W
n

d
d
u

u
d
u

p

Figure 3.27: Feynman diagram of a neutrino CCQE interaction.

(a) νµ(b) ν̄µFigure 3.28: CC cross-section predicted by NEUT version 5.3.2 for (a) νµ and (b)ν̄µ . The T2K flux spectra are shown with arbitrary units (Figures taken from [26]).

3.4. Neutrino event generator NEUT

67

going nucleon has to escape the nuclear potential and experience the Pauli blocking
effects that forces its momentum to be greater than the Fermi surface momentum.
Alternative models are later added to improve data-MC agreement. The spectral
function (SF) model in [120] uses the impulse approximation, where the neutrino
interacts with a single nucleon but takes into account the short-range correlations
between nucleons and affects the energy and momenta of outgoing particles. The
random phase approximation (RPA) model in [121] introduces medium polarization
effect that affects the propagation of particle-hole (ph) pair by the long range correlation induced by residual ph excitation.
On the other hand, the cross-sections for NC elastic scattering are related to theCC cross-sections by the following relations[122, 123]:

3.4.2

σ(νp → νp) = 0.153 × σ(νn → e− p),(3.9)σ(ν̄p → ν̄p) = 0.218 × σ(ν̄p → e+ n),(3.10)σ(νn → νn) = 1.5 × σ(νp → νp),(3.11)σ(ν̄n → ν̄n) = 1.0 × σ(ν̄p → ν̄p).

(3.12)

Single meson production

Single pion production is simulated by the Rein and Sehgal’s method[124], in which
the process is split into two parts:

ν + N → l + N ∗,

3.4. Neutrino event generator NEUT

68

N ∗ → π + N 0,
where N and N 0 are the nucleons, N ∗ is the baryon resonance. The cross-sections
are obtained by multiplying the amplitude of each resonance production with theprobability of pion decay. The excited resonances are included up to an invariant
mass of 2 GeV. The pion angular distribution is calculated by the Rein and Sehgal’s
method for the ∆(1232) resonance, while it is assumed to be isotropic in the resonancerest frame for others. The Pauli blocking effect is considered in the baryon decay,
where the knocked-off nucleon is required to have momentum greater than Fermi
surface momentum. The pion-less ∆ decay (∆ resonances being directly absorbed
by nuclei) is also simulated which accounts for 20% of the events.
The cross-sections for single η and K productions can be obtained by substituting
the corresponding decay probability instead.

3.4.3

Deep inelastic scattering (DIS)For DIS,ν + N → l + N 0 + hadrons,
the cross-section is calculated for hadronic system with an invariant mass energy W
greater than 1.3 GeV. To avoid double-counting of the single pion production process,only processes with pion multiplicites greater than 1 are included for 1.3 GeV < W< 2 GeV. The nucleon structure functions are taken from the parton distributions
functions by Glück et al.[125], with corrections in the lower Q2 region by Bodex andYang[126].
The relations between NC and CC cross-sections are determined from the expe-

3.4. Neutrino event generator NEUT

69

rimental results[127, 128]:




0.26,





Eν < 3GeV

σ(νN → νX)= 0.26 + 0.04(Eν /3 − 1), 3GeV ≤ Eν < 6GeVσ(νN → µ− X) 





0.30,
Eν ≥ 6GeV




0.39,
Eν < 3GeVσ(ν̄N → ν̄X)
= 0.39 − 0.02(Eν /3 − 1), 3GeV ≤ Eν < 6GeVσ(ν̄N → µ+ X) 





0.37,
Eν ≥ 6GeV(3.13)(3.14)

Events with W > 2 GeV are simulated by the CERN PYTHIA/JetSet package[129].
The forward-backward asymmetry of pion multiplicity (nFπ /nBπ ) in the center of massframe of the hadronic system is obtained from the BEBC experimental results[130]:nFπ0.35 + 0.41 ln(W2 )=.
nB
0.5 + 0.09 ln(W2 )
π

3.4.4

(3.15)

Coherent pion production

The coherent pion production model is from [131], where the neutrino interacts withthe entire 16 O nucleus as a whole:ν +16 O → l +16 O + π.

The outgoing pion has the same charge as the incoming weak-current, and pion
production is enhanced in the forward direction as the outgoing pion carries awaymost of the incoming neutrino momentum.

3.4. Neutrino event generator NEUT

3.4.5

70

Multi-nucleon (2p2h) interaction

A multi-nucleon neutrino interaction model by Nieves et al.[121, 132] has been recently added to NEUT. This ’2p2h’ (2 particle-2 hole), or in general ’npnh’, model describes multiple nucleon interaction with a single neutrino. The 2p2h event
topology has a charged lepton and no pions, so is indistinguishable from CCQE
interactions and significantly enhances CCQE-like cross-section.
The cross-sections are calculated by a multi-body expansion of the weak propagator in the medium. While the first-order expansion predicts the standard CCQE
interaction, the second-order and higher order terms involve additional nucleons or∆ resonances in the hadronic current. Since pion-less delta decay (PDD) is alsoa multi-nucleon process, it is possible to include it in the 2p2h model. The 2p2h
PDD cross-sections are confirmed to be similar to the NEUT original implementation in Sec. 3.4.2, therefore only the 2p-2h PDD process is turned on to avoid
double-counting.

3.4.6

Final state and secondary interactions

After the primary interaction processes, leptons and hadrons must escape the nuclear matter before they can be observed. The final state interactions (FSI) between
the hadrons and nuclei can dramatically change the observable particles. In addition, hadrons can still undergosecondary interactions (SI) somewhere in the detector
even after leaving the nucleus, making them invisible in detection. Among all particles, the FSI and SI of pions are the most important, because of both its large
production and interaction cross-section. For example, the charged pion produced ina CC1π interaction can be absorbed by the nucleus and not be detected, mimicking

3.4. Neutrino event generator NEUT

71

a CCQE interaction. The neutrino energy reconstruction can be messed up due tothe wrongly-assumed 2-body kinematics.
NEUT simulates the FSI by an intra-nuclear cascade model. To begin with, theinitial position of the neutrino interaction is generated based on the Woods-Saxon
type nucleon density distribution[133]:

ρ(r) =

Z1
ρ0,
A 1 + exp( r−c)a(3.16)

where Z = 8 is the atomic number and A = 16 is the mass number for the 16 O
nucleus, with the average nuclear density ρ0 = 0.48m3π , and density parameters a =0.41 fm and c = 2.69 fm. The reaction products are then propagated “classically” infinite steps in the nuclear medium. Fermi motion and Pauli blocking are considered.
For pion momentum pπ < 500 MeV/c, the quasi-elastic, single charge exchange
and absorption probabilities are calculated by the Oset et al. model[134]. For
pπ > 500 MeV/c, the interaction probability are computed from PDG cross-section
data[135] of π ± scattering off from free proton and deuteron.
Moreover, nucleon-nucleon elastic scattering is considered, which may produce ∆resonances and pion decays. Cross-section measurements from [136] are used, with
the isobar production model by Lindenbaum et al.[137] included.

Chapter 4
Oscillation samples and event
selections at SK
In order to constrain all the interesting parameters in the sterile neutrino mixing model, we have to measure multiple oscillation probabilities at the same time. Luckily
the SK detector is able to construct different oscillation samples based on observed
event topology. Apart from the standard CC oscillation samples, we have also included two new sets of NC samples (2Rπ 0 and NCγ de-excitation) to enhance thesterile sensitivity. Below we outline all the oscillation samples that are used in thissterile analysis and their selection criteria.

4.1

Date set summary

In this analysis, we make use of data collected during T2K beam periods Runs 1-8.
The duration, proton-on-target (POT) and horn current of each run are summarizedin Table 4.1. The POT for physics analysis only includes good beam spills selected72

4.1. Date set summary
T2K RunStart
End
POT(×1020 )Run 1
23/Jan. 2010 26/Jun. 2010
0.326
Run 2
18/Nov. 2010 11/Mar. 2011
1.122
Run 3b 08/Mar. 2012 21/Mar. 2012
0.217
Run 3c08/Apr. 2012 09/Jun. 2012
1.382
Run 4
19/Oct. 2012 08/May. 2013
3.596
Run 5
16/May. 2014 24/Jun. 2014 0.245/0.512
Run 630/Oct. 2014 03/Jun. 2015 0.192/3.547
Run 7
01/Feb. 2016 27/May. 2016 0.485/3.499
Run 8
27/Oct. 2016 12/Apr. 2017
7.170

73
Horn current set (kA)
+250
+250
+200
+250
+250
+250/-250+250/-250+250/-250+250

Table 4.1: The run period, POT and horn current of each T2K run. Run 5 - Run 7
included periods of beam operation in both neutrino and antineutrino mode. (Table
taken from [27]).
by the SK DAQ status and a number of data quality parameters. Fig. 4.1 shows
the accumulated number of POT as a function of date. The total data for T2K
oscillation analysis is 22.2914 × 1020 POT, with 14.7341 × 1020 POT in neutrino
mode and 7.5573 × 1020 POT in antineutrino mode.
Fig. 4.2 shows the cumulative number of observed fully contained (FC) events
as a function of POT. The stability of event rates is examined by the KolmogorovSmirnov (KS) test. The largest vertical distance D between the data and constantevent rate hypothesis (red line) in Fig. 4.2 is computed for each plot. For neutrino(antineutrino) running, D is found to be 38.05 (12.10) with a normalization factor of
1/1232 (1/263). The KS probability to obtain values larger than D from statisticalfluctuations is 18.43% (61.08%), which is consistent with constant event rate.
The SK detector stability is continuously monitored. For details see Sec. 3.3.4.
Fig. 4.3 shows the time variation of the attenuation length of detector water measured
by cosmic ray through-going muon data, which has remained constant within 90 ± 5
m. The absolute energy scale is also stable, with the energy scale uncertainty be
2.1%.

4.1. Date set summary

74

Figure 4.1: Accumulated number of POT as a function of date for beam good spills(blue), SK + beam good spills in neutrino mode (red), SK + beam good spills in
antineutrino mode (orange), and the SK dead fraction of POT (green = (blue-redorange)/blue) (Figure taken from [27]).

(a) FHC(b) RHC

Figure 4.2: The cumulative number of observed FC events as a function of POT
in neutrino mode (FHC) and antineutrino mode (RHC) for T2K Run 1-8. The redsolid lines refer to expectation in case of constant event rates for all runs (Figures
taken from [27]).

4.2. Charged current oscillation samples

75

Figure 4.3: Time variation of the SK attenuation length measured by cosmic ray
through-going muon data, Grat shaded regions denote the Run 1-8 periods (Figure
taken from [27]).
The MC neutrino events are generated with NEUT 5.3.2, SKDETSIM 13.90 andreweighted by flux 13av1.2. Both the data and MC are reconstructed by apfit version
14b and fiTQun v4r0.

4.2

Charged current oscillation samples

T2K has several standard CC oscillation samples that are sensitive to the 3-flavor
oscillation parameters. In the 3+1 sterile neutrino model that we are interested in,
the CC samples can also put constraints on θ24 . All these samples have been switchedto fiTQun-based reconstruction selection.

4.2.1

FHC/RHC 1Rµ samples

The 1-ring µ-like samples (1Rµ) are the standard T2K νµ disappearance samples
that are available in both neutrino (FHC) and anti-neutrino (RHC) beam modes.

4.2. Charged current oscillation samples

76

The event selection criteria are listed below:
1. Fully-contained (FC) in the SK inner detector (ID) and reconstructed insidethefiducial volume (FV):(a) nhitac < 16 (number of PMT hits in the largest outer detector hit cluster)(b) wall > 50 cm (closest distance between vertex and detector wall),(c) towall > 250 cm (particle track length in ID).
2. Number of rings found by fiTQun multi-ring fitter is one.
3. The ring is identified as muon-like by the PID algorithm.
4. Reconstructed momentum > 200 MeV/c.
5. Number of decay electrons is less than or equal to one.
6. Pass the two-dimensional π + cut.
Figs. 4.4 and 4.5 show the distributions of the variables used in the selection
cuts 2 to 6. Data of T2K Run 1-8 are overlaid on the MC expectation, which show
very good agreement. The numbers of events passing each selection cut are shownin Fig. 4.6. After all selection cuts, 240 events are selected as νµ candidate for FHC
and 68 events for RHC.
Fig. 4.7 show the 1D distributions of the variables wall and towall, and the spatial
distributions of event vertices in R2 and Z of the detector coordinates in Fig. 4.8.
The event rate increases with R from the tank center to the tank wall, due to theincrease in interaction volume, and decreases close to the detector wall due to poorer
reconstruction efficiency. The event rate is flat in the Z direction, except the tank
top/bottom regions.

150

Run1-8 Data600

(14.73×1020 POT)ν µ CC QE
ν µ CC QE
ν µ+ ν µ CC non-QE
ν e + ν e CC
MC w/ T2K+DB bestfit(7.56×1020 POT)ν µ CC QE
ν µ CC QE
ν µ+ ν µ CC non-QE
ν e + ν e CCNC
MC w/ T2K+DB bestfit

50

200

0

Run5-7 Data

100

NC

400

77

Number of events

Number of events

4.2. Charged current oscillation samples

1

2

3

0

≥5

4

1

2

Run1-8 Data
(14.73×1020 POT)ν µ CC QE

60

3

≥5

4

Number of rings
Number of events

Number of events

Number of rings

ν µ CC QE
ν µ+ ν µ CC non-QE
ν e + ν e CCNC
MC w/ T2K+DB bestfit

Run5-7 Data
(7.56×1020 POT)ν µ CC QE
ν µ CC QE

20ν µ+ ν µ CC non-QE
ν e + ν e CCNC
MC w/ T2K+DB bestfit

40

10
20

80

-1000

0
PID parameter

0

1000

Number of events/100 MeV/c

Number of events/100 MeV/c

0

Run1-8 Data
(14.73×1020 POT)ν µ CC QE
ν µ CC QE
ν µ+ ν µ CC non-QE

60ν e + ν e CCNC
MC w/ T2K+DB bestfit

40

-1000

0
PID parameter

1000

Run5-7 Data
(7.56×1020 POT)ν µ CC QE
ν µ CC QE

10ν µ+ ν µ CC non-QE
ν e + ν e CCNC
MC w/ T2K+DB bestfit

5

20

0

0

1000
2000
Momentum (MeV/c)

3000

0

0

1000
2000
Momentum (MeV/c)

3000

Figure 4.4: Distributions of FHC (left) and RHC (right) 1Rµ event selections at eachstage for Run 1-8. Blue arrows denote the selection criteria. The MC expectation is
calculated based on 3-flavor oscillation framework with the world best-fit oscillation
parameters and BANFF fit corrections (Figures taken from [27]).

Run1-8 Data300

(14.73×1020 POT)ν µ CC QE
ν µ CC QE
ν µ+ ν µ CC non-QE
ν e + ν e CC
NC

200

78

Number of events

Number of events

4.2. Charged current oscillation samples

MC w/ T2K+DB bestfit

Run5-7 Data
(7.56×1020 POT)ν µ CC QE

80

ν µ CC QE
ν µ+ ν µ CC non-QE
ν e + ν e CC

60

NC
MC w/ T2K+DB bestfit

40
100
20

0

0

1

2

3

0

≥5

4

0

1

Number of decay-e

2

3

≥5

4

Number of decay-e

300

CC

π

NC

0.6

200

0.5

Data

0.24

CC

0.22

NC

π

Data

log(L +/Lµ)log(L +/Lµ)

300

0.2

200

0.18
0.16

0.4

0.14

100

100

0.12

0.3

0.1
0.08

0.2

0

0

0.06
0.04

0.1

0.02

Number of events

0

1000
2000
3000
reconstructed µ momentum (MeV)0

Run1-8 Data
(14.73×1020 POT)ν µ CC QE

40

-100

0

Number of events-100ν µ CC QE
ν µ+ ν µ CC non-QE
ν e + ν e CCNC
MC w/ T2K+DB bestfit

1000
2000
3000
reconstructed µ momentum (MeV)0

Run5-7 Data
(7.56×1020 POT)ν µ CC QE
ν µ CC QE

10ν µ+ ν µ CC non-QE
ν e + ν e CCNC
MC w/ T2K+DB bestfit

20
5

0

-300

-200

-100
µ/π+ PID

0

100

0

-300

-200

-100
µ/π+ PID

0100

Figure 4.5: Distributions of FHC (left) and RHC (right) 1Rµ event selections at
each stage for Run 1-8. Middle graphs show the π + rejection cut in 2D, with events
below the dotted yellow line selected as νµ candidates, and bottom graphs show the
distance from the cut line (Figures taken from [27]).

1500

Run1-8 Data
(14.73×1020 POT)ν µ CC QE
ν µ CC QE
ν µ+ ν µ CC non-QE
ν e + ν e CC

1000

NC
MC w/ T2K+DB bestfit

500

0

79

Number of events

Number of events

4.2. Charged current oscillation samples

Run5-7 Data
(7.56×1020 POT)ν µ CC QE

300

ν µ CC QE
ν µ+ ν µ CC non-QE
ν e + ν e CCNC

200

MC w/ T2K+DB bestfit

100

FCFV

1-ringµ -likepµ(a) FHC

decay-e

! π+

0

FCFV

1-ringµ -likepµdecay-e! π+(b) RHC

Figure 4.6: Number of events passing each 1Rµ selection stage for T2K Runs 1-8.
The MC expectation is calculated based on 3-flavor oscillation framework with theworld best-fit oscillation parameters and BANFF fit corrections (Figures taken from[27]).
Fig. 4.9 shows the cumulative number of observed νµ events as a function of POT.
For FHC (RHC) data, the greatest distance D between observation and a constantevent rate hypothesis is 0.028 (0.091). The KS probability to obtain large values
due to statistical fluctuations is 98.85% (59.2%). The event rates are stable over theexperimental periods.
From the reconstructed muon momentum pµ , and the angle between the neutrino beam and the outgoing muon θbeam , the reconstructed neutrino energy Eνrec is
calculated byEνrec =

2
(Mn − Vnuc ) · Eµ − m2µ /2 + Mn · Vnuc − Vnuc
/2 + (Mp2 − Mn2 )/2,
Mn − Vnuc − Eµ + pµ cos(θbeam )(4.1)

where Mn (Mp ) is the neutron(proton) mass, Vnuc = 27 MeV is the binding energy ofRun1-8 Data40

(14.73×1020 POT)ν µ CC QE
ν µ CC QE
ν µ+ ν µ CC non-QE
ν e + ν e CC

30

NC
MC w/ T2K+DB bestfit

80

Number of events

Number of events

4.2. Charged current oscillation samples

Run5-7 Data
(7.56×1020 POT)ν µ CC QE

15

ν µ CC QE
ν µ+ ν µ CC non-QE
ν e + ν e CCNC
MC w/ T2K+DB bestfit

10
20
5
10

Number of events

0

Run1-8 Data40

(14.73×1020 POT)ν µ CC QE
ν µ CC QE
ν µ+ ν µ CC non-QE
ν e + ν e CC30

0

200 400 600 800 1000 12001400 1600
wall (cm)NC
MC w/ T2K+DB bestfit

0

Number of events

015

200 400 600 800 1000 12001400 1600
wall (cm)Run5-7 Data
(7.56×1020 POT)ν µ CC QE
ν µ CC QE
ν µ+ ν µ CC non-QE
ν e + ν e CC

10NC
MC w/ T2K+DB bestfit

20
5
10

0

0 500100015002000250030003500400045005000
towall (cm)00 500100015002000250030003500400045005000
towall (cm)Figure 4.7: Reconstructed vertex distributions of FHC (left) and RHC (right) 1Rµevents for Run 1-8, after passing all selection cuts. Top graphs show the wall distribution with towall > 250 cm applied, and bottom graphs show the towall distribution
with wall > 50 cm applied. The blue arrows indicating the full FV cut (Figures
taken from [27]).
neutron in oxygen, mµ is the muon mass and Eµ is its total energy.
In the 1Rµ samples, the main contribution is from νµ (ν̄µ ) charged current quasielastic (CCQE) events with no pion observed in the final state. Fig. 4.10 shows
the reconstructed neutrino energy distribution for T2K Runs 1-8, with data overlaid
on MC. The biggest background is from CC non-QE interactions such as 2p2h andCC1π. There are also contributions from the wrong sign backgrounds, especially in
the RHC data (νµ interaction in RHC mode and vice versa).
The high statistics disappearance samples give us strong sensitivity on sin2 θ23
and ∆m231 . At the same time they also constrain sin2 θ24 as a function ∆m241 .

4.2. Charged current oscillation samples

81

Figure 4.8: Reconstructed vertex distributions of FHC (left) and RHC (right) 1Rµevents for Run 1-8, after passing all selection cuts. Top graphs show the square of
the distance of the vertices from the central vertical axis of the SK tank, and bottom
graphs show the vertical position of the vertices (Figures taken from [27]).

(a) FHC(b) RHC

Figure 4.9: The cumulative number of observed 1Rµ events (blue) as a function of
POT in FHC and RHC for T2K Run 1-8. The red solid lines refer to expectation incase of constant event rates for all runs (Figures taken from [27]).

Run1-8 Data40

(14.73×1020 POT)ν µ CC QE
ν µ CC QE
ν µ+ ν µ CC non-QE
ν e + ν e CC

30

82

Number of events/100 MeV

Number of events/100 MeV

4.2. Charged current oscillation samples

NC
MC w/ T2K+DB bestfit

20

Run5-7 Data
(7.56×1020 POT)ν µ CC QE

10ν µ CC QE
ν µ+ ν µ CC non-QE
ν e + ν e CCNC
MC w/ T2K+DB bestfit

5

10

0

0

1000
2000
Reconstructed ν energy (MeV)

3000(a) FHC

0

0

1000
2000
Reconstructed ν energy (MeV)

3000(b) RHC

Figure 4.10: Reconstructed neutrino energy distribution of 1Rµ samples for T2K
Runs 1-8 (Figures taken from [27]).

4.2.2

FHC/RHC 1Re samples

The 1-ring e-like (1Re) samples are the standard T2K νe appearance samples thatare available in both neutrino (FHC) and anti-neutrino (RHC) beam mode. The
event selection criteria are listed below:
1. FCFV condition:(a) nhitac < 16,(b) wall > 80 cm,(c) towall > 170 cm.
2. Number of rings found by fiTQun multi-ring fitter is one.
3. The ring is identified as electron-like by the PID algorithm.
4. Visible energy Evis > 100 MeV.

4.2. Charged current oscillation samples

83

5. No decay electrons.
6. Reconstructed neutrino energy Eνrec < 1250 MeV.
7. Pass the π 0 rejection cut.
Eνrec is calculated by replacing the muon mass mµ with electron mass me in Eq. 4.1,
and Evis is the sum of momentum of each ring.
Figs. 4.11 and 4.12 show the distributions of the variables used in the selection
cuts 2 to 7. Data of T2K Run 1-8 are overlaid on the MC expectation, which show
very good agreement. The numbers of events passing each selection cut are shownin Fig. 4.13. After all selection cuts, 74 events are selected as νµ candidate for FHC
and 7 events for RHC.
Fig. 4.14 show the 1D distributions of the variables wall and towall, and Fig. 4.15
shows the spatial distributions of event vertex in R2 and Z of the detector coordinates.
Fig. 4.16 shows the cumulative number of observed νe events as a function of
POT. For FHC (RHC) data, the greatest distance D between observation and aconstant event rate hypothesis is 0.099 (0.143). The KS probability to obtain large
values due to statistical fluctuations is 43.8% (99.4%). The event rates are stable
over the experimental periods.
Fig. 4.17 shows the reconstructed neutrino energy distribution for T2K Runs 1-8.
The 1Re samples have a high purity of oscillated νe (ν̄e ) events. The biggest background comes from the intrinsic νe /ν̄e components in the beam. Another importantbackground is due to NCπ 0 events where only one gamma from the π 0 → γγ decay
is reconstructed and wrongly classified as νe events.
The samples are optimized to achieve best sensitivity on the conventional δCP .

Run1-8 Data
(14.73×1020 POT)600

Run5-7 Data
(7.56×1020 POT)

150

Osc. ν e CC
Osc. ν e CCν µ/ ν µ CC
Beam ν e / ν e CCNC
MC w/ T2K+DB bestfit

400

84

Number of events

Number of events

4.2. Charged current oscillation samples

Osc. ν e CC
Osc. ν e CCν µ/ ν µ CC
Beam ν e / ν e CCNC

100

MC w/ T2K+DB bestfit

50

200

0

1

2

3

0

≥5

4

1

2

80

Run1-8 Data
(14.73×1020 POT)
Osc. ν e CC
Osc. ν e CCν µ/ ν µ CC

60

3

≥5

4

Number of rings
Number of events

Number of events

Number of rings

Beam ν e / ν e CCNC
MC w/ T2K+DB bestfit

Run5-7 Data
(7.56×1020 POT)
Osc. ν e CC30Osc. ν e CCν µ/ ν µ CC
Beam ν e / ν e CCNC
MC w/ T2K+DB bestfit

20

40

10
20

-1000

0
PID parameter

0

1000

Run1-8 Data
(14.73×1020 POT)
Osc. ν e CC

20Osc. ν e CCν µ/ ν µ CC
Beam ν e / ν e CC

15NC
MC w/ T2K+DB bestfit

10

Number of events/100 MeV

Number of events/100 MeV

0

-1000

0
PID parameter

6

1000

Run5-7 Data
(7.56×1020 POT)
Osc. ν e CC
Osc. ν e CCν µ/ ν µ CC
Beam ν e / ν e CC

4NC
MC w/ T2K+DB bestfit

2
5

1000
2000
Visible energy (MeV)

Number of events

0

Run1-8 Data150

(14.73×1020 POT)
Osc. ν e CC
Osc. ν e CCν µ/ ν µ CC
Beam ν e / ν e CC
NC

100

0

3000

MC w/ T2K+DB bestfit

0

Number of events

0

1000
2000
Visible energy (MeV)

3000

Run5-7 Data
(7.56×1020 POT)
Osc. ν e CC
Osc. ν e CC20ν µ/ ν µ CC
Beam ν e / ν e CCNC
MC w/ T2K+DB bestfit

10
50

0

0

1

2

3

Number of decay-e

4

≥5

0

0

1

2

3

4

≥5

Number of decay-e

Figure 4.11: Distributions of FHC (left) and RHC (right) 1Re event selections at
each stage for Run 1-8 (Figures taken from [27]).

Run1-8 Data
(14.73×1020 POT)
Osc. ν e CC
Osc. ν e CCν µ/ ν µ CC

40

85

Number of events/250 MeV

Number of events/250 MeV

4.2. Charged current oscillation samples

Beam ν e / ν e CCNC
MC w/ T2K+DB bestfit

20

Run5-7 Data
(7.56×1020 POT)
Osc. ν e CC
Osc. ν e CC6ν µ/ ν µ CC
Beam ν e / ν e CCNC
MC w/ T2K+DB bestfit

4

2

Data
CC
NC

400

0

3000

0.3
0.25

0

1000
2000
Reconstructed ν energy (MeV)

3000

Data
CC

400

0.14
0.12
0.1

0.2
200

0.16

NC

π

1000
2000
Reconstructed ν energy (MeV)

πlog(L 0/Le)

0log(L 0/Le)

0

0.08

200

0.15

0.06
0.1

0.04

0.05
0

100
200
300
reconstructed π0 mass (MeV)

Run1-8 Data
(14.73×1020 POT)
Osc. ν e CC
Osc. ν e CC20ν µ/ ν µ CC
Beam ν e / ν e CCNC
MC w/ T2K+DB bestfit

0

Number of events

Number of events

0

0.02

0
0

100
200
300
reconstructed π0 mass (MeV)0

Run5-7 Data
(7.56×1020 POT)
Osc. ν e CC

3Osc. ν e CCν µ/ ν µ CC
Beam ν e / ν e CCNC
MC w/ T2K+DB bestfit

2
10
1

0

-100

0
100
200
e/π0 PID parameter

300

0

-100

0
100
200
e/π0 PID parameter

300

Figure 4.12: Distributions of FHC (left) and RHC (right) 1Re event selections at
each stage for Run 1-8. Middle graphs show the π 0 rejection cut in 2D, with events
below the dotted yellow line selected as νe candidates, and bottom graphs show the
distance from the cut line (Figures taken from [27]).

Run1-8 Data
(14.73×1020 POT)

1500

Osc. ν e CCν µ/ ν µ CC
Beam ν e / ν e CCNC
MC w/ T2K+DB bestfit

500

0Run5-7 Data
(7.56×1020 POT)
Osc. ν e CC

300

Osc. ν e CC

1000

86

Number of events

Number of events

4.2. Charged current oscillation samples

Osc. ν e CCν µ/ ν µ CC
Beam ν e / ν e CCNC
MC w/ T2K+DB bestfit

200

100

FCFV

1-ringe-likeevis

decay-eEνrecfiTQun0

FCFV

1-ring(a) FHCe-likeevis

decay-eEνrecfiTQun(b) RHC

Figure 4.13: Number of events passing each 1Re selection stage for T2K Runs 1-8(Figures taken from [27]).
On the other hand, they also put some constraints on θ24 .

FHC νe CC1π + sample

4.2.3

The νe CC1π + sample focuses on selecting CC1π + instead of CCQE events. The
Feynman diagram of CC1π + interaction is shown in Fig. 4.18. The π + is searched byisidentifying its decay electron in event. The reconstructed neutrino energy Eνrece CC∆given by:
Eνrec
=
e CC∆

2Mp Ee + M∆2 ++ − Mp2 − Me2,
2(Mp − Ee + pe cos θbeam )(4.2)

with M∆++ = ∆++ resonance mass (1232.0 MeV/c2 ).
The selection criteria of CC1π + sample are thus very similar to 1Re samples,
except that it requires exactly one decay electron reconstructed in event. Currently
only neutrino mode data of Runs 1-8 is available for this sample.
1. FCFV condition:(a) nhitac < 16,Run1-8 Data
(14.73×1020 POT)
Osc. ν e CC
Osc. ν e CC20ν µ/ ν µ CC
Beam ν e / ν e CCNC
MC w/ T2K+DB bestfit

87

Number of events

Number of events

4.2. Charged current oscillation samples

6

Run5-7 Data
(7.56×1020 POT)
Osc. ν e CC
Osc. ν e CCν µ/ ν µ CC
Beam ν e / ν e CC

4NC
MC w/ T2K+DB bestfit

102

Number of events

0

500

1000
wall (cm)

20

0

1500

0

Run1-8 Data
(14.73×1020 POT)
Osc. ν e CC
Osc. ν e CCν µ/ ν µ CC

15

Beam ν e / ν e CCNC
MC w/ T2K+DB bestfit

Number of events

0

500

1000
wall (cm)3

1500

Run5-7 Data
(7.56×1020 POT)
Osc. ν e CC
Osc. ν e CCν µ/ ν µ CC
Beam ν e / ν e CC

2NC
MC w/ T2K+DB bestfit

10

1
5

0

0

1000

2000
3000
Towall (cm)

4000

5000

0

0

1000

2000
3000
Towall (cm)

4000

5000

Figure 4.14: Reconstructed vertex distributions of FHC (left) and RHC (right) 1Re
events for Run 1-8, after passing all selection cuts. Top graphs show the wall distribution with towall > 170 cm applied, and bottom graphs show the towall distribution
with wall > 80 cm applied. The blue arrows indicating the full FV cut (Figures
taken from [27]).
(b) wall > 50 cm,(c) towall > 270 cm.
2. Number of rings found by fiTQun multi-ring fitter is one.
3. The ring is identified as electron-like by the PID algorithm.
4. Visible energy Evis > 100 MeV.
5. One decay electron.
6. Reconstructed neutrino energy Eνrec< 1250 MeV.
e CC∆

4.2. Charged current oscillation samples

88

Figure 4.15: Reconstructed vertex distributions of FHC (left) and RHC (right) 1Re
events for Run 1-8, after passing all selection cuts. Top graphs show the square of
the distance of the vertices from the central vertical axis of the SK tank, and bottom
graphs show the vertical position of the vertices (Figures taken from [27]).

(a) FHC(b) RHC

Figure 4.16: The cumulative number of observed 1Re events (blue) as a function of
POT in FHC and RHC for T2K Run 1-8. The red solid lines refer to expectation incase of constant event rates for all runs (Figures taken from [27]).

Run1-8 Data
(14.73×1020 POT)
Osc. ν e CC
Osc. ν e CCν µ/ ν µ CC

20

Beam ν e / ν e CCNC
MC w/ T2K+DB bestfit

10

89

Number of events/125 MeV

Number of events/125 MeV

4.2. Charged current oscillation samples

3

Run5-7 Data
(7.56×1020 POT)
Osc. ν e CC
Osc. ν e CCν µ/ ν µ CC
Beam ν e / ν e CC

2NC
MC w/ T2K+DB bestfit

1

0

0

0

500
1000
Reconstructed ν energy (MeV)

0

500
1000
Reconstructed ν energy (MeV)(a) FHC(b) RHC

Figure 4.17: Reconstructed neutrino energy distribution of 1Re samples for T2K
Runs 1-8 (Figures taken from [27]).

νl

l
W

p

d
u
u

−

π+

u
u
u
u

d
++

Δ

d
u
u

p

Figure 4.18: Feynman diagram of a CC1π + interaction
7. Pass the π 0 rejection cut.
The distributions of variables from the decay electron cut are shown in Fig. 4.19.
Fig. 4.20 shows the number of events passing each selection cut. After all selection
cuts, 15 events are found.
Fig. 4.21 and 4.22 show the distributions of the 1D vertex variables.
The reconstructed neutrino energy distribution is shown in Fig. 4.23. As an
extension to the 1Re samples, the νe CC1π + sample is also a very pure oscillated νe

Run1-8 Data150

(14.73×1020 POT)
Osc. ν e CC
Osc. ν e CCν µ/ ν µ CC
Beam ν e / ν e CC
NC

100

90

Number of events/250 MeV

Number of events

4.3. Neutral current oscillation samples

MC w/ T2K+DB bestfit

Run1-8 Data
(14.73×1020 POT)
Osc. ν e CC
Osc. ν e CCν µ/ ν µ CC

10Beam ν e / ν e CCNC
MC w/ T2K+DB bestfit

5

50

0

0

1

2

3

0

≥5

4

0

CC

400

0.025

NC

0.02

0.015

Number of events

Data

π

log(L 0/Le)

Number of decay-e

1000
2000
Reconstructed ν energy (MeV)

3000

Run1-8 Data
(14.73×1020 POT)
Osc. ν e CC

4Osc. ν e CCν µ/ ν µ CC
Beam ν e / ν e CC

3NC
MC w/ T2K+DB bestfit

200
0.01

0.005

2

1

0

0

100
200
300
reconstructed π0 mass (MeV)

0
0

-100

0
100
200
e/π0 PID parameter

300

Figure 4.19: Distributions of FHC νe CC1π + event selections at each stage for Run
1-8. Bottom left graph shows the π 0 rejection cut in 2D, with events below the dottedyellow line selected as νe candidates, and bottom right graph shows the distance fromthe cut line(Figures taken from [27]).
CC interaction sample, although it has lower statistics.

4.3

Neutral current oscillation samples

Apart from the CC oscillation samples mentioned above, SK also provides few NC
oscillation samples are that sensitive to θ24 and θ34 . While these NC samples are
already published before, this is the first time to use them in oscillation analysis.

91

Number of events

4.3. Neutral current oscillation samples

Run1-8 Data
(14.73×1020 POT)

1500

Osc. ν e CC
Osc. ν e CCν µ/ ν µ CC
Beam ν e / ν e CC
NC

1000

MC w/ T2K+DB bestfit

500

0

FCFV

1-ringe-likeevis

decay-eEνrecfiTQunRun1-8 Data
(14.73×1020 POT)
Osc. ν e CC

4Osc. ν e CCν µ/ ν µ CC
Beam ν e / ν e CC

3NC
MC w/ T2K+DB bestfit

Number of events

Number of events

Figure 4.20: Number of events passing each FHC νe CC1π + selection stage for T2K
Runs 1-8 (Figures taken from [27]).

6

Run1-8 Data
(14.73×1020 POT)
Osc. ν e CC
Osc. ν e CCν µ/ ν µ CC
Beam ν e / ν e CC

4NC
MC w/ T2K+DB bestfit

2
2
1

0

0

500

1000
wall (cm)

1500

0

0

1000

2000
3000
Towall (cm)

4000

5000

Figure 4.21: Reconstructed vertex distributions of FHC νe CC1π + events for Run 1-8,
after passing all selection cuts. Left graph shows the wall distribution with towall >270 cm applied, and right graph shows the towall distribution with wall > 50 cm
applied. The blue arrows indicating the full FV cut (Figures taken from [27]).

4.3.1

FHC/RHC 2Rπ 0 samples

The 2-ring pi-zero (2Rπ 0 ) sample events are reconstructed by apfit instead of fiTQun.
This sample searches for neutral current (NC) interaction events with a single π 0
production. Data are available in both neutrino and antineutrino beam mode. The
selection criteria are as follows:

4.3. Neutral current oscillation samples

92

Number of events/125 MeV

Figure 4.22: Reconstructed vertex distributions of FHC (left) and RHC (right) 1Re
events for Run 1-8, after passing all selection cuts. Top graphs show the square of
the distance of the vertices from the central vertical axis of the SK tank, and bottom
graphs show the vertical position of the vertices (Figures taken from [27]).

Run1-8 Data
(14.73×1020 POT)
Osc. ν e CC
Osc. ν e CC6ν µ/ ν µ CC
Beam ν e / ν e CCNC
MC w/ T2K+DB bestfit

4

2

0

0

500
1000
Reconstructed ν energy (MeV)

Figure 4.23: Reconstructed neutrino energy distribution of FHC νe CC1π + sample
for Runs 1-8 (Figures taken from [27]).
1. FCFV condition:(a) nhitac < 16,(b) wall > 200 cm,(c) towall > 200 cm.
2. Number of rings found by apfit is two.

4.3. Neutral current oscillation samples

93

3. Both rings are identified as electron-like by the PID algorithm.
4. No decay electrons.
5. Reconstructed invariant mass 85 MeV < minv < 185 MeV.
Fig. 4.24 shows the number of ring distribution for all FCFV multi-ring events.
The two-ring selection contains quite a lot of CC and NC non-1π backgrounds.
The subsequent cuts 3-5 successfully remove most background with a high selection
efficiency of NC1π events. After all selection cuts, 53 events are found in FHC data,
and 9 events in RHC data.

(a) FHC(b) RHC

Figure 4.24: Number of ring distribution of all apfit FCFV multi-ring events for Runs
1-8 (Figures taken from [27]).

From MC simulations, 88% of events originate from NC interactions where asingle π 0 is produced, either from ∆ resonance or coherent pion production, anddecays into 2γ. If we count all NC events, the overall NC purity reaches over 97%.
Figs. 4.25 and 4.26 shows the reconstructed π 0 invariant mass, reconstructed π 0momentum and true neutrino energy distributions.

4.3. Neutral current oscillation samples

94(a) Reconstructed π 0 invariant mass (b) Reconstructed π 0 invariant massfor FHC.
for RHC.

(c) Reconstructed π 0 momentum for (d) Reconstructed π 0 momentum for
FHC.
RHC.

Figure 4.25: Reconstructed π 0 invariant mass and momentum of 2Rπ 0 samples forRuns 1-8 (Figures taken from [27]).
The NC 2Rπ 0 samples are actually useful in both 3-flavor and sterile oscillation
analysis. For the 3-flavor analysis, the 2Rπ 0 samples can help constraining π 0 background in the 1Re samples. On the other hand, any deficit in the NC samples might
be regarded as the direct evidence of active-sterile neutrino oscillation.
Fig. 4.26 shows the true neutrino energy of the 2Rπ 0 events, which spans over
the whole T2K beam energy range. One important difference compared with the

8

CC

7

NC 1π

6

NC other

595

Events/7.56× 1020 POT

Events/14.30× 1020 POT

4.3. Neutral current oscillation samples

CC

1.4

NC 1π

1.2

NC other
1
0.8

4
0.6

3

0.4

2

0.2

1
0
0

1

2

3

4

5

0
0

6
7
8
9
10
True neutrino energy (GeV)(a) True neutrino energy for FHC.

1

2

3

4

5

6
7
8
9
10
True neutrino energy (GeV)(b) True neutrino energy for RHC.

2
True neutrino energy (GeV)

True neutrino energy (GeV)

Figure 4.26: True neutrino energy distribution of 2Rπ 0 samples.

1.8
1.6
1.4
1.2
1
0.8
0.6

1.6
1.4
1.2
1
0.8
0.6

0.4

0.4

0.2
0

2
1.8

0.2
90

100

110

120

130

140

150 160 170 180
π invariant mass (MeV)
0

0
0

100

200

300

400

500

600

700 800 900 1000
π0 momentum (MeV)(a) True neutrino energy vs. π 0 invariant(b) True neutrino energy vs. π 0 momentum.
mass.

Figure 4.27: Correlations between neutrino energy and π 0 invariant mass/momentum
for FHC.

4.3. Neutral current oscillation samples

96

CC samples is that, we can only reconstruct the π 0 kinematics, which has little
information about the neutrino energy. Fig. 4.27 shows, e.g. the correlations between
reconstructed π 0 invariant mass/momentum and true neutrino energy. The 2Rπ 0samples are thus sensitive to θ24 and θ34 but not ∆m241 .

4.3.2

FHC NCγ de-excitation sample

This is another NC oscillation sample that searches forneutral current quasi-elastic(NCQE) interaction. In a neutrino-nucleus (16 O) NCQE interaction, usually a single
nucleon is knock out:ν +16 O → ν +15 N ∗ + p,
ν +16 O → ν +15 O∗ + n.
The excited 15 N or 15 O atom emits a primary gamma of around 6 MeV, while theknocked-out nucleon (proton or neutron) further interacts to emit secondary gammas.
These low energy events are reconstructed using special techniques instead of thestandard algorithm apfit or fiTQun.

NCQE cross-section
The NCγ de-excitation MC are generated separately from other samples. In thesimulation, the total NCQE cross-sections σν 16 O are calculated by the impulse approximation and spectral function (SF):X Z
M d2 σνNd2 σν 16 O
=
dpdEP
(p,
E)
,
N
dΩdEν0EN dΩdEν0
N =p,n(4.3)

4.3. Neutral current oscillation samples

97

where Eν0 is the energy of the outgoing neutrino, EN =p
M 2 + p2 , and M = 0.938

GeV is the nucleon mass. The total cross-section is written as an incoherent sum of2d σνNthe individual cross-sections on free nucleons dΩdE
0 , weighted by the spectral functionνPN (p, E).
The spectral function PN (p, E) defines the probability of removing a nucleon of
momentum p and leaving the residual nucleus with energy E + E0 − M , where E0
is the ground state energy. In the nuclear shell model, the spectral function can be
written as
PN (p, E) =X

ni |φi (p)|2 fi (E − Ei ),

(4.4)

i

where φi (p) and Ei are the wave function and the binding energy of the i-th state
respectively. The occupation probability ni and the probability density functionfi (E − Ei ) take into accounts the nucleon-nucleon (N N ) correlations. Without N N
correlations, ni = 1 and fi (E − Ei ) = δ(E − Ei ).

Figure 4.28: The binding energy spectra of 16 O spectral function. The two peaks
correspibd to the 1p1/2 -hole state (ground state) and 1p3/2 -hole state respectively.
Neutron levels are deeper than protons’ by about 3.54 MeV (Figure taken from[28]).
The oxygen spectral function is adopted from Benhar et al.[138, 139], calculatedby the local density approximation (LDA)[140] with data from (e, e0 p) experiments[141].

4.3. Neutral current oscillation samples
Residual nucleus
15
N
15
O

1p1/2 1p3/2
1s1/2
0.158 0.3515 0.1055
0.158 0.3515 0.1055

98
others0.385
0.385

Table 4.2: The spectroscopic factors of each nucleon state in 16 O (Table taken from[28]).
Fig. 4.28 shows the energy spectra obtained by integrating PN (p, E) over p. The
1p1/2 and 1p3/2 states have sharp peaks but the 1s1/2 state is spread over very highenergy due to N N correlations. The neutron and proton levels have the same shapewith energy shifted by 3.54 MeV.
The probability of finding the residual nucleus in a specific shell state, called
the spectroscopic factor, is obtained by integrating the energy spectra in Fig. 4.28.
The factors for 1p1/2 , 1p3/2 and 1s1/2 are integrated between 11.0 ≤ E ≤ 14.0 MeV,
17.25 ≤ E ≤ 22.75 MeV and 22.75 ≤ E ≤ 62.25 MeV respectively. Table 4.2
summarizes the values for residual nuclei 15 N (proton-hole states) and 15 O (neutronhole states).

Nuclear de-excitation
Following the NCQE nucleon knock-out, the residual nucleus may be excited andemit de-excitation gamma rays (and other particles) afterwards. The de-excitation
branching ratios are mostly obtained from experimental data.
The 1p1/2 is the ground state, so no decay happens.
For the 1p3/2 proton hole state, there are three possible energy levels: 6.32 MeV,
9.93 MeV and 10.70 MeV. The branching ratios are assigned according to the experimental data of the 16 O(e, e0 p)15 N[142]. The de-excitation gamma branching ratios
are taken from [143]. The 6.32 MeV level only emits single gamma ray, while the 9.93

4.3. Neutral current oscillation samples

Residual
nucleus15
N

15

Energy level
(MeV)
6.32
9.93

O

10.70
6.18
9.61
10.48

Jiπ

Ratio

3−
2
3−
2

0.869
0.049

3−
2
3−
2
3−
2
3−
2

0.082
0.869
0.049
0.082

99

Eγ
Ep
Ratio
(MeV)
(MeV)
6.32
−
1.0
9.93
−
0.776±0.019
5.27+5.30−
0.154±0.015
6.32−
0.049±0.012
7.30−
0.021±0.008
−
0.5
−
6.18
−
1.0
−
0.5
−
−
0.5
−

Br(Xi∗ → γ + Y )
0.869
0.038
0.008
0.002
0.001
−
0.869
−
−

Table 4.3: Summary of the energy levels and gamma ray emission branching ratios
for the 1p3/2 hole states (Table taken from [28]).
MeV level has several emission channels. The 10.70 MeV does not emit gamma but afree nucleon instead. For the 1p3/2 neutron hole state, there is no experimental dataso the spectroscopic factors are assigned to be same as that of proton hole. However,
only the 6.18 MeV level emits gamma ray at 86.9% branching ratio. Table 4.3 lists
the branching ratio of the 1p3/2 hole states.
For the 1s1/2 proton hole state (15 N∗ ), the branching ratios are measured by theE148 experiment on 16 O(p, 2p)15 N reactions at the Research Center for Nuclear Physics (RCNP)[144–146]. The neutron-hole state (15 O∗ ), again, has not been measured
intensively yet, so the branching ratios are assigned the same as that of the proton
hole state. Since the energy levels of the 1s1/2 hole state are higher than the nucleon
emission level, usually a nuclear fragmentation happens instead of gamma emission.
The decay particles are further simulated to check whether there is de-excitation
gamma. Fig. 4.29 shows, e.g. the branching ratios of 15 N∗ for gamma rays with
Eγ > 3 MeV.

4.3. Neutral current oscillation samples

100

Figure 4.29: De-excitation modes of the 16 O proton hole (s1/2 )−1
p state. Only gamma
rays with Eγ > 3 MeV are shown (Figure taken from [29]).

4.3. Neutral current oscillation samples

101

Event reconstruction
To reconstruct a NCQE event, the SK low energy analysis tools[147] are used. The
fitter, BONSAI (branch optimization navigating successive annealing iterations), performs a maximum likelihood fit to each vertex candidate against both Cherenkov
signal and dark noise background hypotheses. The reconstructed energy thresholdis set as 4 MeV which is the same as SK-IV solar neutrino analysis. The energy
spectrum of signal events has a peak at around 6 MeV, so the 4 MeV threshold hasa trigger efficiency above 99.5%. An upper limit of 30 MeV is set to eliminate thedecay electron background.
Next, a timing cut is applied to select “in-timing” events that are within ±100
ns from the beam bunch centers. A sample of “off-timing” background is also constructed by selecting events that are 5 to 500 µsec before each neutrino spill arriving
edge, and is used for the estimation of beam-unrelated background.
After that, a series of wall distance cuts and BONSAI fit goodness cuts are applied
to remove the radioactivity background. Since most backgrounds appear near the
ID wall, the vertex wall distance cuts should remove most of them. A FV cut isapplied to reject events with vertices < 2 m away from the ID wall, and then twowall variables are defined: dwall is the distance from the reconstructed vertex to thenearest ID wall, and effwall is the distance from the reconstructed vertex along the
backward of the reconstructed direction to the ID wall.
The vertex goodness parameter GV and direction goodness parameter GA fromBONSAI fit are used to remove events with mis-reconstructed vertices. GV is definedby the timing information of the PMT hits:

GV =1 ∆ti 2− ()
ie 2 σ
iw
P
,
i wi

P

(4.5)

4.3. Neutral current oscillation samples

102

with ∆ti be the PMT hit residual time:∆ti = ti −|~hi − ~x|
,
c0

(4.6)

where ti is the PMT time, ~hi is the PMT position, ~x is the reconstructed vertex and
c0 is the light velocity in water. Ideally we have ∆ti ≈ 0 for all PMT hits for perfectvertex reconstruction. σ = 3.0 ns is the characteristic PMT timing resolution for asingle photo-electron. The weighting factor wi of the i-th hit PMT is used to reduce
accidental dark noise:
1 ∆ti 2wi = e− 2 ( σw ) ,(4.7)

where σw = 60 ns is a much wider effective time resolution. A large GV implies that∆ti has narrow distribution and the vertex is reconstructed very precisely.
The direction (angular) goodness GA assesses the azimuthal uniformity of the
PMT hits relative to the reconstructed direction. It uses the KS test to calculate themaximum distance between the PMT data and uniform azimuthal emission hypothesis:GA =max{∠uni (i) − ∠data (i)}
,
2π

(4.8)

where ∠data (i) is the azimuthal angle of the i-th PMT hit, ∠uni (i) = N2π50 × i is theangle of uniform azimuthal emission, and N50 is the number of PMT hits within a
50 ns time window.
The total event fit goodness is defined asovaQ≡ G2V −G2A . Fig. 4.30 shows the G2V
vs. G2A distributions for simulated beam events and beam-unrelated events which aremostly radioactive background. A cut at, e.g. ovaQ = 0.2 can effectively separate
the signal from background events.

4.3. Neutral current oscillation samples

103

Figure 4.30: Distributions of G2V vs. G2A of NCγ de-excitation events with reconstructed energy between 4.5 and 5.0 MeV. The left plot shows the distribution ofT2K beam νµ simulated events, while the right plot shows that of the off-timing
data. The dotted lines show a cut at ovaQ≡ G2V − G2A = 0.2 (Figures taken from[28]).
The cut values of dwall, effwall and ovaQ are optimized from run to run based on
SK detector conditions, e.g. radioactive impurities of ID wall and water transparency,
and the T2K beam power. This is done by maximizing the following figure-of-meritover the cut parameters:

F OM = p

Nsignal.
Nsignal + Nbackground(4.9)

Fig. 4.31 shows, e.g. the cut thresholds for T2K Run 4.
Finally, an event pre-activity cut is applied 0.2-20 µs before the event to reject
low energy cosmic or atmospheric neutrino muons and their decay electrons.Only
events with N30 (maximum number of PMT hits in 30 ns time window) less than 22
are accepted. A Cherenkov angle cut of θC > 34o is used to reject low energy muonor pion that are produced in beam-related CC and NC others interactions, since they
have smaller Cherenkov opening angle.
Fig. 4.32 shows the reconstructed gamma energy distributions of event candidates
before and after the selection cuts for T2K Run 4. The selection efficiency of beam

4.3. Neutral current oscillation samples

104

Figure 4.31: Cut threshold of dwall, effwall and ovaQ as a function of reconstructedenergy for T2K Run4. Red points show the optimized cut values in each energy bin,while red dash lines show the fitting lines (Figures taken from [28]).
events is 89.6%, and the rejection rate of beam-unrelated background is more than
99.9%.

(a) Beam MC events and off-timing
data.

(b) Run 4 data.

Figure 4.32: Distributions of reconstructed energy of NCγ de-excitation events forT2K Run 4 before and after selection cuts (Figures taken from [28]).

Fig. 4.33 shows the reconstructed gamma energy and true neutrino energy distributions of the NCγ de-excitation sample for T2K Run 1-4. The biggest backgroundin this sample is the NC1π 0 production from ∆ resonance, where the π 0 is absorbed
in final state interaction and the excited nucleus emits de-excitation gamma similar

105

Events/6.56× 1020 POT

4.3. Neutral current oscillation samples

22

NCQE

20

NC non-QE

18

CC

16
14
12
10
8
6
4
2
0
0

1

2

3

4

5

6
7
8
9
10
True neutrino energy (GeV)(b) True neutrino energy.
(a) Reconstructed gamma energy.

Figure 4.33: Distributions of FHC NCγ de-excitation sample for Runs 1-4.
to that in NCQE interaction. There are also a small amount of CC interaction backgrounds due to charged lepton production with a low momentum. The lepton might
be mis-identified as a γ-ray event due to mis-constructed Cherenkov angle. Or, whena nucleon knock-out happens in a CCQE interaction, primary de-excitation γ-ray isemitted from the excited nucleus, and secondary γ-rays are produced by the freenucleon.
Currently only neutrino beam mode data from T2K Run 1-4 data are processed,and 102 events are found. From MC simulations, around 70% of events are fromNCQE interactions, 25% from other NC interactions, and the rest are from CCinteractions and beam-unrelated background.
Fig. 4.34 shows the correlation between reconstructed gamma energy and true
neutrino energy. Again the reconstructed gamma energy distribution has no information about incoming neutrino energy, so it has no sensitivity on ∆m241 .

106

True neutrino energy (GeV)

4.4. Event summary

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

5

10

15

20
25
30
Reconstructed γ energy (MeV)

Figure 4.34: Correlation between reconstructed gamma energy and true neutrinoenergy of FHC NCγ de-excitation sample.

4.4

Event summary

Table 4.4 summarizes the number of events for each CC and NC sample.
SampleBeam mode
No. of events

1Rµ
FHC RHC
240
68

1Re
FHC RHC74
7

νe CC1π +
FHC
15

2Rπ 0
FHC RHC
53
9

NCγ de-excitation
FHC
102

Table 4.4: Number of events observed in each CC and NC oscillation samples.

Chapter 5
Oscillation analysis framework
The objective of our study is to constrain the sterile mixing parameters. We do thisby performing a joint oscillation analysis with all the SK samples discussed above.
While we only explicitly use SK data, many of the systematic parameters are actuallyconstrained by the BANFF fit with the near detector data and propagated to thefar detector. Below we describe our analysis strategy and the systematic parameters
applied to this analysis.

5.1

Event binning

In order to maximize the sensitivity to various oscillation parameters, events indifferent oscillation samples are binned uniquely.
For the 1Rµ samples, events are binned one-dimensionally in reconstructed energyEνrec . The binning is dense in the first few GeV in order to map out accurately the
first oscillation minimum at 0.6 GeV. The event rate is much lower in the highenergy region due to the off-axis, narrow beam energy spectrum, and the bin width107

5.2. Monte Carlo PredictionsRange
0 - 3 GeV
3 - 4 GeV
4 - 6 GeV
6 - 10 GeV
10 - 30 GeV

108Size of one bin Number of bins
0.05 GeV
60
0.25 GeV
4
0.5 GeV
4
1 GeV
420 GeV
1

Table 5.1: Binning used for reconstructed energy of 1Rµ samples.
is therefore larger.
For the 1Re and νe CC1π + samples, events are binned two-dimensionally in electron momentum pe and scattering angle θbeam with respect to the beam. The p-θ
distribution is different between signal and background categories, hence providing
additional discrimination power between signal and background events.
For the 2Rπ 0 and FHC NCγ de-excitation examples, events are binned in π 0momentum and reconstructed gamma energy respectively. However, since the event
distributions have little dependence on neutrino energy, these samples mostly matterin the overall rate but not shape.
The full binning scheme for all samples is shown in Table 5.1 - 5.4.

5.2

Monte Carlo Predictions

The oscillation fit mainly works by comparing real data with MC expectations assuming certain oscillation parameters. The MC used for prediction is generated byfeeding NEUT neutrino interaction products into SKDETSIM, then processing theresults by the standard reconstruction algorithm and storing the neutrino events indedicated format. Some important features are discussed here.

5.2. Monte Carlo Predictions

109

Range
Size of one bin Number of bins
0 - 1500 MeV/c 100 MeV/c
15
0 - 140 degree
10 degree
14
140 - 180 degree 40 degree
1
180
160
140
120
100
80
60
40
20
0
0

200

400

600

800

1000

1200

1400

Table 5.2: Binning used for reconstructed lepton momentum and angle of 1Re samples and FHC νe CC1π sample.
Range
0 - 1500 MeV/cSize of one bin Number of bins
100 MeV/c
15

Table 5.3: Binning used for reconstructed π 0 momentum of 2Rπ 0 samples.

5.2.1

Neutrino propagation modes

Neutrino events are classified into six types of neutrino propagation modes: νµ → νµ ,
ν̄µ → ν̄µ , νe → νe , ν̄e → ν̄e , νµ → νe , ν̄µ → ν̄e . ντ (ν̄τ ) oscillations are ignored
because of the relatively high energy CC interaction threshold. νe → νµ and ν̄e → ν̄µtransitions are also negligible. Since all neutrino flavors undergo NC interactions at
the same rate, we only consider NC events in νµ → νµ and ν̄µ → ν̄µ modes.
Range
4 - 30 MeV

Size of one bin Number of bins
1 MeV
26

Table 5.4: Binning used for reconstructed γ energy of FHC NCγ de-excitation sample.

5.2. Monte Carlo Predictions

5.2.2

110

Neutrino interaction categories

In MC files, NEUT neutrino interaction modes are stored for each neutrino event.
For analysis purpose, we group the NEUT modes into nine categories: chargedcurrent quasi-elastic interactions (CCQE), 2p2h, charged-current interactions associated with single-pion resonant production (CC1π), charged-current coherent pion
production (CC coherent), other charged-current processes (CC other), neutral current interactions with single pion production (NC1π), NC coherent pion production(NC coherent), NC1γ (NC1γ), and other neutral current interactions (NC other).
Table 5.5 shows the relationship between the NEUT modes and the nine interaction
categories.
NEUT interaction mode
1
2
11-13
16
17-29
31-34
36
38,39
37,40-52

Category in analysis
CCQE
2p2h
CC1π
CC coherent
CC other
NC1π
NC coherent
NC1γ
NC other

Table 5.5: Definition of neutrino interaction categories in this analysis. NEUT interaction modes of anti-neutrinos are negative. NCQE events (NEUT mode 51,52)
are classified as NC other.

5.2.3

True neutrino energy binning

The true neutrino energy Eνtrue has 84 bins. Again it has more bins around theoscillation minimum at 0.6 GeV (see Table. 5.6).

5.2. Monte Carlo Predictions
Range
0 - 0.3 GeV
0.3 - 1 GeV
1 - 3 GeV
3 - 3.5 GeV
3.5 - 4 GeV
4 - 5 GeV
5 - 7 GeV
7 - 10 GeV
10 - 30 GeV

111
Size of one bin Number of bins
0.05 GeV
6
0.025 GeV
28
0.05 GeV
40
0.1 GeV5
0.5 GeV
1
1 GeV
1
2 GeV
1
3 GeV
1
20 GeV
1

Table 5.6: Binning used for true neutrino energy Eνtrue .

5.2.4

Input preparation

To reduce the computation time for an oscillation fit, MC neutrino events are grouped
into histograms based on propagation modes, interaction categories and true energies
as described above. In each histogram, events are binned according to sample type
as in Sec. 5.1. In order to correctly account for the event contribution, different
weights are applied to the MC events.
First, the near detector measures accurately the neutrino beam flux as a functionof neutrino flavors and energy. The run to run beam power variations are taken into
account to provide an overall weight at a certain beam energy.
The second weight is related to the neutrino interaction model. The default
CCQE MC events are generated using the spectral function (SF) cross-section model,
while later studies found that the use of relativistic Fermi gas (RFG) model andthe
Bernstein parameterization of random phase approximation (BeRPA) gives better
agreement with external data. Therefore a reweighting is applied to correct thecross-section weight (see Sec. 5.3.2).
Finally, two extra weights are applied: true NC1γ events are given double weightbased on recent findings [148], and CC coherent pion events (NEUT code 16) have

5.2. Monte Carlo Predictions

112
Eπ (GeV) Weight0-0.250.135
0.25-0.5
0.4
0.5-0.75
0.294
0.75-1.0
1.206

Table 5.7: Weights applied to CC coherent pion events.
been tuned as a function of the pion energy to the Berger-Sehgal model [149] following
recent MINERνA results (see Table 5.7).

5.2.5

Oscillation probability calculation

The standard analysis uses the Prob3++[150] software package for probability calculation. The package employs the Barger et al.[151] model to calculate the full
three-flavor oscillation probability analytically. Matter effect in the Earth is included with the radial density structure given by the PREM model[152].
For the sterile analysis, the vacuum “3+1” oscillation probability without approximation in Sec. 2.2.3 is added, assuming eV scale sterile neutrino with ∆m241 . 1eV2 .
Also, with four mass eigenstates, the matter effect included Hamiltonian (similar to
Eq. 2.13) cannot be diagonalized analytically, so numerical methods has to be employed. However, for T2K beam energy and baseline, matter effect has little impacts
on the oscillation spectra. We simply use the vacuum probability in this study.

5.2.6

Event rate calculation

In the oscillation analysis, the expected event rates of all samples can be calculated
for a specific set of oscillation parameters and systematic parameters. To do this,5.2. Monte Carlo Predictions

113

the input produced in Sec. 5.2.4 is first loaded:

Mb (C, I, Eνtrue ),(5.1)

where b is the sample-specific event bin, C is the neutrino propagation mode, I is
the interaction mode, Eνtrue is the true neutrino energy.
Oscillation probabilities Posc (C, I, Eνtrue ; o) are then applied to produce the event
templates Tb (C, I, Eνtrue ; o):Tb (C, I, Eνtrue ; o) = Mb (C, I, Eνtrue )Posc (C, I, Eνtrue ; o),
(−)

(−)

(−)

(5.2)

(−)

where o are the oscillation parameters. For ν µ → ν µ mode, P (ν µ → ν µ ) is applied
(−)

(−)

on CC interactions and 1 − P (ν µ → ν s ) is applied on NC interactions:
(−)

(−)

(−)

(−)

Posc (ν µ → ν µ , CC, Eνtrue ; o) = P (ν µ → ν µ )≈ 1 − sin2 2θ23 cos 2θ24 sin22∆m231 L2
2 ∆m41 L
−
sin
2θ
sin
,
24
4Eνtrue
4Eνtrue

(5.3)

(−)

(−)

(−)

(−)

Posc (ν µ → ν µ , NC, Eνtrue ; o) = 1 − P (ν µ → ν s )≈ 1 − sin2 θ34 sin2 2θ23 sin22∆m231 L2
2
2 ∆m41 L
−
cos
θ
sin
2θ
sin
.
34
24
4Eνtrue
4Eνtrue

(5.4)
(−)

(−)

(−)

(−)

For ν e → ν e mode, P (ν e → ν e ) is applied on CC interactions:
(−)

(−)

(−)

(−)

Posc (ν e → ν e , CC, Eνtrue ; o) = P (ν e → ν e ) ≈ 1 − sin2 2θ13 sin2

∆m231 L
.
4E

(5.5)

5.3. Systematic parameters
(−)

(−)

(−)

114
(−)

For ν µ → ν e mode, P (ν µ → ν e ) is applied on CC interactions:
(−)

(−)

(−)

(−)

Posc (ν µ → ν e , CC, Eνtrue ; o) = P (ν µ → ν e ) ≈ sin2 2θ13 cos2 θ24 sin2 θ23 sin2

∆m231 L.
4E
(5.6)

In addition, when systematic parameters f are varied from their nominal values,
there will be an extra reweighting factor Rb (C, I, Eνtrue ; f ) applied to the templates.
Therefore the predicted number of event in the bin b is given by

b
Npred
=

X

Tb (C, I, Eνtrue ; o)Rb (C, I, Eνtrue ; f ).

(5.7)

C,I,Eνtrue

5.3

Systematic parameters

In our sterile neutrino analysis, a large number of systematic parameters are implemented. In general, they can be classified into three categories: beam flux, neutrino
interaction and cross-section, and Super-K detector uncertainties. Many of them are
well-constrained by the near detector and devoted SK studies. A covariance matrix
taking into account the correlation of these parameters is input into the analysis
code, and is used in the calculation of likelihood values. Fig. 5.1 shows the error sizeand correlation of all the 133 systematic parameters.

5.3.1

Beam flux parameters

The uncertainties on the neutrino flux are divided into 50 parameters by neutrino
flavor, true energy and beam mode (see Table 5.8 and 5.9). The beam parameters areincorporated into the BANFF framework and the uncertainties are reduced throughconstraints from the near detector data.

5.3. Systematic parameters

Event category
Oscillated νe , νµOscillated ν̄e , ν̄µ
νe
ν̄e

115

Beam ν flavor Energy binning (GeV)# of bins
νµ flavor
0-0.4, 0.4-0.5, 0.5-0.6, 0.6-0.7,
11
0.7-1.0, 1.0-1.5, 1.5-2.5, 2.5-3.5,
3.5-5.0, 5.0-7.0, 7.0-30.0
ν̄µ flavor
0-0.7, 0.7-1.0, 1.0-1.5, 1.5-2.5,
5
2.5-30.0
νe flavor
0-0.5, 0.5-0.7, 0.7-0.8, 0.8-1.5,
7
1.5-2.5, 2.5-4.0, 4.0-30.0
ν̄e flavor
0-2.5, 2.5-30.0
2

Table 5.8: Binning of beam flux systematic parameters for neutrino mode.

Event category
Oscillated νe , νµOscillated ν̄e , ν̄µνeν̄eBeam ν flavor Energy binning (GeV)# of bins
νµ flavor
0-0.7, 0.7-1.0, 1.0-1.5, 1.5-2.5,
5
2.5-30.0ν̄µ flavor
0-0.4, 0.4-0.5, 0.5-0.6, 0.6-0.7,
11
0.7-1.0, 1.0-1.5, 1.5-2.5, 2.5-3.5,
3.5-5.0, 5.0-7.0, 7.0-30.0
νe flavor
0-2.5, 2.5-30.0
2ν̄e flavor
0-0.5, 0.5-0.7, 0.7-0.8, 0.8-1.5,
7
1.5-2.5, 2.5-4.0, 4.0-30.0

Table 5.9: Binning of beam flux systematic parameters for anti-neutrino mode.

116

Parameter Bin

Fractional error

5.3. Systematic parameters

1.4
1.2
1

1
120

0.8
0.6

100

0.4

80

0.2

0.8

0

60

-0.2

0.6

40

0.4

-0.6

20

0.2
0
0

-0.4

20

40

60

80

0
0

100
120
Parameter Bin

-0.8
20

40

60

80

100

120

-1

Parameter Bin(a) Error (square root of diagonal terms)(b) Uncertainties correlation matrix

Figure 5.1: Error size and correlations for all the 133 systematic parameters of theanalysis. Parameters are 1-50: beam flux, 51-72: neutrino interaction and crosssection, 73-132: Super-K detector, 133: SK p-scale parameters.

5.3.2

Neutrino interaction and cross-section parameters

The BANFF fit has parameterized 20 parameters to describe the uncertainties inneutrino interaction model. Although all these parameters are included in the neardetector fit, not all receive a significant constraint. On the other hand, we haveadded two non-BANFF parameters to take care the cross-section normalization error
for NC1π and NCQE interactions. A full list of these parameters can be found in
Table 5.10. The type “shape” means its effect is energy-dependent, while “norm” isenergy-independent normalization parameter.
As mentioned in Sec. 5.2.4, CCQE interactions are described by the RFG modelwith relativistic RPA applied. The two fundamental parameters, axial mass MAQE
and Fermi momentum pF on oxygen, are allowed to vary. The axial mass entersthe formula of CCQE cross-section for free nucleon in the form of axial vector form
factor:FA = gA

Q2
1+(MAQE )2

!−2
,

(5.8)

5.3. Systematic parameters

117

Parameterinteraction categories applied

type

Nominal value & uncertainty

MAQE

CCQE
CCQE
2p2h
2p2h
2p2h
2p2h
CCπ and NC1π
CCπ and NC1π
CCπ and NC1π
CCCC
CC other
CC coherent
NC coherent
NC 1γ
NC other
CCQE
CCQE
CCQE
CCQE
NC1π
NC other#

shape
shape
norm
norm
norm
shape
shape
shape
shape
norm∗
norm∗
shape
norm
norm
norm∗
norm∗
shape
shape
shape
shape
normˆ
normˆ

1.13±0.08 (GeV/c2 )204.98±15.08 (MeV/c)
1.50±0.20
0.73±0.23
0.96±0.17
1.00±0.35
0.98±0.06
0.81±0.04 (GeV/c2 )1.31±0.26
1.00±0.03
1.00±0.03
0.39±0.20
0.87±0.28
0.94±0.30
1.00±1.00
1.00±0.30
0.69±0.06
1.60±0.12
0.96±0.13
0.87±0.35
1.00±0.30
1.00±0.30

pF
2p2h norm. ν
2p2h norm. ν̄
2p2h norm. C to O
2p2h shape O
CARES
MARES
BGRES
A
σνe /σνµσν̄e /σν̄µCC other shape
CC coherent
NC coherent
NC 1γ
NC other
BeRPA ABeRPA BBeRPA CBeRPA D
NC1π
NCQE

Table 5.10: Summary of the neutrino interaction parameters using the results of
ND280 fit. ∗ Errors are not constrained by the near detector. ˆ Non-BANFF errors
added for the sterile analysis. # NCQE error instead of NC other error applies for
FHC NCγ de-excitation sample.

5.3. Systematic parameters

118

where Q2 is the four-momentum transfer and gA = −1.267 is the axial-vector weakcoupling constant. The Fermi momentum is a parameter in the Fermi gas model,
where the knocked-out nucleon needs to have large enough momentum to escape thenucleus.
Since it is found that one correction term for relativistic RPA is not flexible
enough, an effective RPA function (BeRPA) is developed to have the same Q2 dependence as the Nieves RPA model[121] and cover the theoretical errors at the sametime. The parameterization is given as

f (x) =
A(1 − x0 )3 + 3B(1 − x0 )2 x0 + 3(C − 1)(1 − x0 )x02 + Cx03 , x < U1 + (C + U D(C−1) ) exp(−D(x − U )),
3

(5.9)

x>U

where x = Q2 , x0 = x/U . Five parameters named A, B, C, D and U allows variationof various Q2 region of the RPA function but U = 1.2 GeV2 is fixed in the analysis.
A nominal BeRPA weight is assigned in the cross-section weight, and the variationfrom the nominal value is taken care by the four parameters.
The 2p2h interactions have four parameters. There is one overall normalization
parameter for ν and ν̄ interactions respectively. A normalization parameter (2p2h
norm. C to O) of 20% is introduced to take into account the difference between
carbon and oxygen. A shape parameter is introduced to test the effects of artificiallyredistributing the 2p2h cross section strength between “non-Delta like” to “Deltalike” 2p2h model.
For CC1π and NC1π interactions, there are systematic parameters that change
the scaling factor for the axial form factor CA5 , the resonant axial vector mass MARES ,and the isospin-1/2 (I1/2 ) continuum background (BGRES). CA5 affects the scale ofA

5.3. Systematic parameters

119

the axial form factor at Q2 = 0, and MARES affects both the rate and Q2 shape ofinteractions. BGRES
is responsible for the scaling of the non-resonance process in I1/2
A
interaction channels. In addition MiniBooNE and MINERνA data, the parameters
are also tuned against bubble chamber experiments ANL[153, 154] and BNL[155–157]
to provide priors for BANFF.
There are separated normalization parameters for CC coherent, NC coherent,
NC1γ and NC other interactions, and shape parameter for CC other interactions.
The cross-section ratio uncertainties σνe /σνµ and σν̄e /σν̄µ are applied to νe and ν̄eevents, because only νµ and ν̄µ cross-sections are directly constrained by the neardetector fit.
In our sterile analysis, the 2Rπ 0 and NCγ de-excitation samples contain mostly
NC1π and NCQE events respectively (NCQE interaction is classified into NC otherinteraction category), therefore two relevant cross-section normalization parameters
are added. They are not constrained by the BANFF fit, and we determined to assign
them a conservative value of 30% because there are no accurate data constraining
the cross-section values.
The NC1π normalization error is set as the same size as NC coherent normalization error, where these two parameters are usually correlated in measurements. The
value of 30% is the also same as in previous oscillation analysis[158, 159]. Comparison of NEUT with experimental data such as MiniBooNE[160] shows that it is areasonable value. The T2K off-axis Pi-zero detector (PØD) is designed to measure
the NC1π 0 events in water, and we expect the cross-section error can be reducedwith more data in the future.
For the NCQE interactions, there is publication that estimates the theoreticaluncertainty of NCQE cross-sections by comparing the cross-section values calculated

5.3. Systematic parameters
Neutrino flavor
Osc. νe /ν̄e
νµ /ν̄µBeam νe /ν̄e
All

Interaction mode
CC
CC
CC
NC

120
Momentum Binning (GeV/c)
0-0.3, 0.3-0.7, 0.7-1.5
0-0.3, 0.3-0.7, 0.7-1.5
0-0.3, 0.3-0.7, 0.7-1.5
0-0.3, 0.3-0.7, 0.7-1.5

Table 5.11: Binning of SK detector systematic parameters for 1Re and νe CC1π
samples.
by different models[161]. Among them, the biggest difference is around 30%, between
spectral function (SF) and relativistic Green’s functions (RGF). The same author has
also published results that compare SF cross-section with several experimental data
sets [162]. The reported difference is around 20% when compared to MiniBooNE,
but the measurement is on carbon but not oxygen, so we may need to scale up theerror. Based on above findings, we determined to set the error as 30%. MiniBooNE
has also reported measurements on NCQE/CCQE cross-section ratio on CH2 [163],
but the reported errors are quite large that it does not help much in reducing ourerror size. There is analysis going on within T2K to measure the NCQE cross-section
with ND280 FGD, and we might expect error reduction by then.

5.3.3

Super-K detector uncertainties

The SK detector uncertainties contain detector efficiency uncertainties, final stateinteraction (FSI) and secondary interaction (SI) model uncertainties,photo-nuclear(PN) effect uncertainties, and energy scale uncertainty. All these uncertainties (except energy scale) are combined by summing the corresponding covariance matrices.
Events are divided into different modes and bins based on sample type. This gives
us a total of 12 × 3 + 6 × 2 + 4 × 2 + 4 = 60 bins which are summarized in Table 5.11
- 5.14.

5.3. Systematic parameters

Neutrino flavorνµ /ν̄µνµ /ν̄µνe /ν̄eAll

121

Interaction mode
CCQE
CC OtherCC
NC

Energy Binning (GeV/c)
0-0.4, 0.4-1.1, 1.1-30.0
0-30.00-30.00-30.0

Table 5.12: Binning of SK detector systematic parameters for 1Rµ samples.

Neutrino flavor
Osc. νe /ν̄e
νµ /ν̄µBeam νe /ν̄e
All

Interaction mode
CC
CC
CC
NC

Table 5.13: Binning of SK detector systematic parameters for 2Rπ 0 samples.

Neutrino flavor
Interaction modeAll
NC other (NCQE)All
All NC interactions except NC otherAll
CC
Beam unrel.
Table 5.14: Binning of SK detector systematic parameters for NCγ de-excitation
sample. Beam unrel. refers to background events that are not due to neutrino
interactions, e.g. PMT dark noises, radioactive decay, etc.

5.3. Systematic parameters
Event Typeνe CC1eνe CC otherνµ CC1µνµ CC
νµ CCµπ 0 otherNC 1π 0NC π 0 otherNC γ
NC 1π ±
NC other122
Selection criteria
νe CC && Nπ0 ==0 && Nπ± ==0 && NP ==0νe CC && !(νe CC1e)
νµ CC && Nπ0 ==0 && Nπ± ==0 && NP ==0νµ CC && Nπ0 ==0 && !(νµ CC1µ)νµ CC && Nπ0 >0
NC && !(NC γ) && Nπ0 ==1 && Nπ± ==0 &&NP ==0
NC && !(NC γ) && Nπ0 ≥1 && !(NC 1π 0 )NEUT mode == 38 k NEUT mode == 39
NC && !(NC γ) && Nπ0 ==0 && Nπ± ==1 &&NP ==0
NC && !(NC 1π 0 ) && !(NC π 0 other) &&!(NCγ) && !(NC 1π ± )Table 5.15: Criteria for event categorization based on final state information. The
number of charged pions (Nπ± ) and protons (NP ) only includes particles producedwith momentum above Cherenkov threshold set at 156.0 MeV/c and 1051.0 MeV/c
respectively.
SK detector efficiency uncertainties
For all samples except NCγ de-excitation, the SK detector efficiency uncertainties arecalculated by evaluating the fractional change of the number of events and correlation
between event categories when the selection cut parameters are varied with a ToyMC
method. These uncertainties describe the mis-identification of the observed event
topologies (final state FS) with designed topological cuts. The FS modes are listed
in Table 5.15. The errors for different FS modes are evaluated by cosmic muon
control samples, atmospheric neutrino samples and hybrid-π 0 samples.

Cosmic muon control samples The cosmic muon samples are to used assess
both vertex and decay electron cut uncertainties. For vertex uncertainty, the wall
distributions of the entering cosmic muon events are studied. We know the true
wall value should be 0 cm, so the reconstructed wall values give the vertex residual

5.3. Systematic parameters

Tagging efficiency (ηtag )Decay electron cut

123
Data
84.7 ± 0.2 %0.21 ± 0.02 %Reweighted MC
Uncertainty
86.0 ± 0.2 %1%−3(9.5 ± 0.6)×10 %0.2%

Table 5.16: Decay electron efficiency and estimated systematic uncertainties.
distribution. Comparing between data and MC in Fig. 5.2, a 2.5 cm difference isobserved. Simulations are done to shift the reconstructed wall values by ±2.5 cm tosee the change in the number of events in each sample, and less than 0.5% uncertainty
is observed. The decay electron cut uncertainty is studied by the stopping muon
sample that is reweighed to better match T2K’s kinematics. The tagging efficiency
and fake rate are evaluated for MC and data (see Fig. 5.3), and the uncertainties are
summarized in Table 5.16. The uncertainty for νe samples is calculated by attributing
both the fake rate systematic of events without a true decay electron and the tagging
efficiency of events with a true decay electron:2
σdecay-e= (Ptrue decay-e × σtag )2 + ((1 − Ptrue decay-e ) × σfake )2 ,(5.10)

where σtag and σfake are the values given in Table 5.16, and this gives a systematic
uncertainty of 0.2% for both beam and signal νe samples. Furthermore, the systematic uncertainty on the mis-identification (mis-ID) of muons as electrons is estimated.
The muon mis-ID rate is defined asηµ−misID =1dcye
Ne-like
1dcye
1dcye
Ne-like+ Nµ-like,(5.11)

1dcye
1dcye
where Ne-like(Nµ-like) is the number of e-like (µ-like) events with 1 decay electron.

The muon mis-ID rate is measured to be 3.9 ± 0.2% for data and 3.0 ± 0.2% forreweighted MC (see Fig. 5.4), so the systematic uncertainty is around 30%.

5.3. Systematic parameters

124

Figure 5.2: The normalized wall distributions of stopping cosmic muons for MC
simulation (red) and data (black) (Figures taken from [30]).

(a) Tagging efficiency.

(b) Fake rate.

Figure 5.3: Decay electron tagging efficiency and fake rate as a function of muon
momentum (left) and towall (right) (Figures taken from [30]).
Atmospheric neutrino fit The atmospheric neutrino data in SK is used to estimate the systematic parameters dSK related to the fiTQun reconstruction performance. The dSK parameters are varied to maximize the likelihood between MC5.3. Systematic parameters

125

Figure 5.4: Muon mis-identification rate for stopping muon data and MC, as afunction of muon momentum (left) and towall (right) (Figures taken from [30]).
predictions and SK atmospheric data:

L(a, xatm , dSK |MSK ) = P (MSK |a, xatm , dSK ) × π(a) × π(xatm ) × π(dSK ), (5.12)

where a are the atmospheric neutrino flux parameters, xatm are the atmospheric neutrino cross-section parameters, MSK is the observed SK atmospheric neutrino data,and π are the parameter prior distributions. dSK contains parameters that modify
the fiTQun topological cuts, listed in Table 5.17. Each parameter is parametrized in
the following way:L0m = β 1 Lm + β 0 ,(5.13)

where β 0 is a bias parameter to the Lm distribution, and β 1 is a smearing parameterthat adjusts the width of the distribution. The flux and cross-section uncertainties
are separated from the detector systematics so that only the detector components arepropagated to the T2K analyses. Also, the detector is divided into six regions by thewall and towall variables (see Fig. 5.5) to allow spatial variation of the reconstruction
efficiency in the detector. The fitting results in each sample bin are specified by the“shift error” which is the fractional deviation of number of events from nominal, and5.3. Systematic parameters
Cut Variable m
0
1
2
3

126
Cut Variable NamefiTQun e/µ PID
fiTQun e/π 0 PID
fiTQun µ/π PID
fiTQun Ring-Counting Parameter

Table 5.17: List of fiTQun cut variables used in the atmospheric neutrino fit.
the “fit error” which represents the uncertainty of measurements. Fig. 5.6 shows theresults of SK topological errors for FS modes of νe CC1e, νe CC other, νµ CC1µ andνµ CC other.

Figure 5.5: Graphical representation of the detector region bin as a function of walland towall, zoomed into small wall and towall regions (Figures taken from [31]).

Hybrid-π 0 fit A set of hybrid-π 0 samples are constructed to evaluate the modellingof π 0 events in SK. The hybrid-π 0 events are built by overlaying a electron-like ringfrom SK atmospheric samples or a decay electron ring from a stopping cosmic ray
muon with a simulated photon ring, where the rings are chosen such that they matchthe momenta and opening angle of the T2K-MC NCπ 0 decay kinematics. They arefurther divided into primary and secondary samples in which the electron constitutes

5.3. Systematic parameters

127

Figure 5.6: Overall SK topological cut errors as a function of visible energy. The
total error in each bin is the sum in quadrature of the “shift error” (bin content) and“fit error” (error bar). Top left: νe CC1e, bottom left: νe CC other, top right: νµ
CC1µ, bottom right: νµ CC other (Figures taken from [31]).
the higher and lower energy rings respectively. The hybrid-π 0 MC samples with bothrings from simulation are compared with the hybrid-π 0 data samples to study thedifference in the fraction of events that pass the νe selection. The relative difference
between data and MC is defined as:R=MC − data,
data(5.14)

where data (MC ) is the remaining efficiency after selections for data (MC). The
statistical error on the ratio is calculated assuming the data and MC are independent:
MC
δR =
×
data

r
(

δMC 2 δdata 2
) +
).
MC
data

(5.15)

5.3. Systematic parameters
Error source
VertexDecay electron cut
Topological cuts128
νe CC1π +1Rµ0.5%
0.1%⊕0.2% 1%⊕2% 0.1%⊕0.2%
Tables 5.19, 5.20, 5.21
1RνeTable 5.18: Input systematic errors for the ToyMC of SK detector efficiency uncertainties.
Event typeRing-counting Particle ID π 0 rejectionνe CC1e
atm-ν fitνe CC otheratm-ν fitνµ CC DIF (19%)
16%νµ CC non-DIF (81%)
36.1%
νµ CC other150%νµ CC π 0 otherfiTQun hybrid-π 0NC 1π 0fiTQun hybrid-π 0NC π 0 otherfiTQun hybrid-π 0NC γνe CC1e+1%
±
NC 1π
100%
100%
100%
NC other
100%
100%
100%
Table 5.19: Input systematic errors of topological cuts (ring-counting, PID, π 0 rejection cuts ) for 1Re and νe CC1π + samples.
The final systematic error is obtained by adding the data/MC differences in the
primary and secondary samples independently in quadrature, i.e.
q

2
2
2
2
Rprimary+ δRprimary+ Rsecondary+ δRsecondary.

(5.16)

Table 5.18 shows the systematic error inputs to the ToyMC. In the ToyMC, each
event is assigned with a weight derived from the random fluctuations of the estimatedsystematic errors. The weighted events are then summed by the output bin and thefractional shift from the nominal value is calculated. The process is repeated by 106
times, and the errors and correlations between the output bins are calculated.

5.3. Systematic parameters
Event typeνµ CC1µνµ CC otherνe CC1µNC

129
Ring-counting Particle ID
atm-ν fit
atm-ν fit100%
atm-ν fit = 64.5%

Table 5.20: Input systematic errors of topological cuts (ring-counting, PID ) for 1Rµsamples.
Event type
NC 1π 0NC π 0 other
Other NC
CC

Topological cut errors
apfit hybrid-π 0 = 6.49%
apfit hybrid-π 0 = 25.37%
100%
100%

Table 5.21: Input systematic errors of topological cuts for 2Rπ 0 samples.
SK detector efficiency uncertainties for NCγ de-excitation sample
For the NCγ de-excitation sample, the SK errors are estimated by separated study
because the event selection processes are very different from other samples. We
are interested in three error sources: primary gamma production, secondary gamma
production and detector responses.
The primary gamma emission from NCQE events are determined by the spectroscopic factors of each shell states and the branching ratios of de-excitation gammas. The
spectroscopic factors control the probability of appearance of the hole states (1p1/2 ,1p3/2 , 1s1/2 ) after single nucleon knock out. Among them, only the 1p3/2 and 1s1/2
states are excited, and their branching ratios of de-excitation gammas are quite precisely measured. For 1p3/2 hole states, the dominant error is from the spectroscopic
factor and branching ratios 9.93 MeV level, which induces a 3% change in event number. The uncertainties of the 1s1/2 states are small, but there are continuum excitedstates above 1s1/2 that are not simulated. They are treated in a similar way as 1s1/2

5.3. Systematic parameters
Sourcecontinuum states
1p3/2
1s1/2
Eγ < 3 MeV
multi-holes
total

130
Uncertainties (%)
3
3
1
1
9
10

Table 5.22: The uncertainties of primary gamma production for NCQE signal events.
since they both have high enough energy compared to particle emission thresholds.
Because more than 70% of 1s1/2 states do not emit gamma, the change in event number is small. Gamma rays with Eγ < 3 MeV are not simulated in NEUT because
there are very few available experimental data, but they can hardly produce highenough energy Compton electrons for Cherenkov emission. The biggest uncertainty,
however, comes from multi-nucleon hole states, because there are no experimental
data about the de-excitation of these multi-hole states. This uncertainty is estimatedby completely switching off gamma emission for these states, and the expected event
rate drops by 9%. Table 5.22 lists the contribution from different sources of errors,
and the overall primary gamma production error is determined to be 10%.
For events due to NC1π interaction, where de-excitation gammas are produced
from π absorption in nucleus, the uncertainty is evaluated by switching the branching ratios from default to two different scenarios: zero gamma emission which
reduces event rates by 0.6%, and branching ratios be the same as NCQE events
which increases event rates by 2.5%. So the uncertainty is set as 3%.
Secondary gamma productions are mainly due to knocked-out neutrons. The neutrons can further interact with oxygen or be captured by hydrogen. Since there are
no experimental constrain on gamma ray emission from neutron-oxygen interaction,
the uncertainties are calculated by comparing the simulation results of different si-

5.3. Systematic parameters

131

SourceUncertainty (%)
Energy scale
0.4
Energy resolution
1
Trigger efficiency
1
dwall
1
effwall
0.2
ovaQ
0.2
Pre-activity< 0.1
Cherenkov angle
1
Total
2.1
Table 5.23: The detector response uncertainties for NCγ de-excitation sample.
mulators. The nominal simulation uses MICAP[164, 165] (NMTC[166]) simulator
for neutrons energy Tn below (above) 20 MeV, while the alternative simulation uses
MICAP (NEUT) for Tn below (above) 30 MeV. In the region Tn > 30 MeV, theaverage energy of secondary gammas from NEUT is higher than that from NMTC,
which changes the expected number of events by 12%.
The detector response uncertainties are those related to event selection cut parameters. Table 5.23 lists all items. Energy scale and energy resolution are calibrated
precisely in SK, and they introduce around 1% change in event number. Trigger
efficiency is evaluated by calibration data and MC, and uncertainty is 1%. The cut
values of parameters dwall, effwall and ovaQ change from run to run for selection
optimization. Shifts in the cut parameters change the event numbers by less than
1% for each item. The pre-activity cut of N30 < 22 is tight, and the uncertainties
of dark noise or low momentum muons that pass the cut are below 0.1%. The Cherenkov angle cut is shifted by the difference of Cherenkov angle distributions fromcalibration source and MC, and the resultant changes of remaining events is 1%.
Table 5.24 summarizes the systematic uncertainties of the NCγ de-excitation
sample. The uncertainty of the number of beam-unrelated background events is just

5.3. Systematic parameters

132

Category
Primary γ production
Secondary γ production
Detector response
Total

NCQE
10%
13%
2.1%
16.5%

Other NC CC
3%
6%
13%
7.6%
2.1%
2.1%
13.5%
9.9%

beam-unrelated
0.8%

Table 5.24: The summary of SK detector efficiency errors for NCγ de-excitation
sample.
statistical uncertainty of the accidental events in the 23 hours off-timing data.
Fig. 5.7 shows the size and correlation of the SK detector efficiency parameters
for all the oscillation samples. Note that the correlations between the CC and NC
samples are weak because they are reconstructed by different algorithm (fiTQun,

Parameter Bin

Fractional error

apfit, BONSAI).
1.4
1.2
1
0.8

60

1
0.8

50

0.6
0.4

40

0.2
30

0
-0.2

0.6
20

-0.4

0.4

0
0

-0.6

10

0.2

-0.8
10

20

30

40

50

60

Parameter Bin(a) Error (square root of diagonal terms)

0
0

10

20

30

40

50

60

-1

Parameter Bin(b) Uncertainties correlation matrix

Figure 5.7: Error size and correlations for the SK detector efficiency uncertainties.
Parameters are 1-12: FHC 1Re, 13-18: FHC 1Rµ, 19-30: RHC 1Re, 31-36: RHC
1Rµ, 37-48: FHC νe CC1π + , 49-52: FHC 2Rπ 0 , 53-56: RHC 2Rπ 0 , 57-60: FHC NCγ
de-excitation.

FSI and SI model uncertainties
The FSI and SI model is parameterized by six scale factors. There are three parameters that scale the NEUT microscopic cascade interaction mechanism probabilities

5.3. Systematic parameters133

for absorption (FSIABS), low energy QE scattering (FSIQE) and the low energy
single charge exchange (SCX) branching fraction (FSICX). Additionally, there arethree parameters chosen to span the errors in the high energy data: QE scattering(FSIQEH), SCX (FSICXH) and pion production (FSIINEL).
Pion scattering data are used to tune and constrain the model parameters. Constrains for the low momentum parameters (FSIABS, FSIQE, FSICX) are obtainedby fitting the NEUT model to π ± -C data[44]. A grid search is performed on thethree parameters. The best fit parameter set is found, and 8 parameter sets from thecorner of each octant of the 3-parameter space that intersects the 1-σ surface of the
fit are chosen to represent the constraints and correlations between the parameters.
For the high energy parameters (FSIQEH, FSICXH, FSIINEL), two parameter
sets are chosen to span the errors in high momentum (up to 1500 MeV/c) external
data. This gives us a total of 8 × 2 = 16 parameter sets to propagate the FSI uncertainties. Table 5.25 shows the nominal and 16 variations of NEUT FSI parameters
for error estimation.
The NEUT cascade implementation in the SK detector simulator SKDETSIM
allows the simultaneous estimation of FSI and SI uncertainties by reweighting thecross-section weights of MC events with the parameters in Table 5.25. The fractionchange of events in each output bin and the correlations are calculated in the formof covariance matrix.

PN model uncertainties
SKDETSIM contains a model of photo-nuclear effect: a γ-ray photon could be absorbed by a nucleus so that it would not be reconstructed. This can lead to a π 0 → γγ
decay being mis-identified as a νe /ν̄e event. A conservative 100% uncertainty is5.3. Systematic parameters

Par. Set FSIQE
Nominal
1.0
1
0.6
2
0.6
3
0.7
4
0.7
5
1.4
6
1.3
7
1.5
8
1.6
23
0.6
24
0.6
25
0.7
26
0.7
27
1.4
28
1.3
29
1.5
30
1.6

FSIQEH
1.8
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
2.3
2.3
2.3
2.3
2.3
2.3
2.3
2.3

134

FSIINEL
1.0
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5

FSIABS
1.1
0.7
0.7
1.6
1.6
0.6
0.7
1.5
1.6
0.7
0.7
1.6
1.6
0.6
0.7
1.5
1.6

FSICX
1.0
0.5
1.6
0.4
1.6
0.6
1.6
0.4
1.6
0.5
1.6
0.4
1.6
0.6
1.6
0.4
1.6

FSICXH
1.8
2.3
2.3
2.3
2.3
2.3
2.3
2.3
2.3
1.3
1.3
1.3
1.3
1.3
1.3
1.3
1.3

Table 5.25: NEUT FSI “1-sigma” parameter sets from the fit to π ± -C data[44].
applied to this type of events in 1Re and νe CC1π + samples.
Fig. 5.8 shows the size and correlation of FSI+SI+PN parameters.

SK energy uncertainty
The SK energy uncertainty is applied after all other nuisance parameters to see how
the sample spectrum changes if we multiply Erec /p of all the events by the energy
scale parameter. We assume that events are uniformly distributed in each bin andthus if a fraction of α% of the energy range covered by the bin i moves into the rangeof bin i+1, that α% of events would behave in the same manner. The final event
rates used in likelihood calculation is produced after this event migration has been
calculated.

135

Parameter Bin

Fractional error

5.3. Systematic parameters

0.25
0.2

60

1
0.8

50

0.6
0.4

40

0.2

0.15
30

0
-0.2

0.1

20

0.05

10

0
0

0
0

-0.4
-0.6
-0.8

10

20

30

40

50

60

10

Parameter Bin(a) Error (square root of diagonal terms)

20

30

40

50

60

-1

Parameter Bin(b) Uncertainties correlation matrix

Figure 5.8: Error size and correlations for FSI+SI+PN uncertainties. Parameters are
1-12: FHC 1Re, 13-18: FHC 1Rµ, 19-30: RHC 1Re, 31-36: RHC 1Rµ, 37-48: FHC
νe CC1π + , 49-52: FHC 2Rπ 0 , 53-56: RHC 2Rπ 0 , 57-60: FHC NCγ de-excitation.

5.3.4

Effect of systematic parameters

In this section we study the effects of systematic parameters on event rates of the eightoscillation samples. For each error category, we make 10k throws of the systematicparameters based on the covariance matrix and calculate the RMS of the predicted
event rates among the throws. The total systematic errors are calculated by making
50k throws of all parameters. Table 5.26 summarizes the error contributions for the
standard CC samples, assuming oscillation parameter set in Table 6.2, and Table 5.27
summarizes that for the newly added NC samples. We can clearly see that the total
errors for CC samples are only a few percent (except νe CC1π + ), while the errors
are over 20% for NC samples due to the larger contribution in cross-section andSK errors. Note that the cross-section errors for the NC samples are roughly 20%,
because the errors are dominated by the NC1π/NCQE normalization parametersthat affects about 70% of events in the samples.
Fig. 5.9, 5.10 and 5.11 show the event distribution variations due to systematicparameters. The distributions are quite symmetric around the nominal values, and5.3. Systematic parameters

Error Source
Beam
Neutrino interaction and cross-section
SK Detector
Syst. Total

136

1Rµ
FHC RHC
4.1
3.8
4.7
4.0
3.3
2.9
4.7
4.0

1Re
FHC RHC
4.3
3.9
5.5
5.2
3.8
4.3
5.8
6.4

νe CC1π +
FHC
4.3
5.0
17.1
17.7

Table 5.26: Percent errors on the predicted event rates of CC samples.

Error Source
Beam
Neutrino interaction and cross-section
SK Detector
Syst. Total

2Rπ 0
FHC RHC
4.2
3.8
20.1 19.1
8.8
8.6
21.3 20.4

NCγ de-excitation
FHC
4.1
21.1
13.2
23.3

Table 5.27: Percent errors on the predicted event rates of NC samples.
the size of error envelops are consistent with the tables above.
To understand better the error contribution, we vary the systematic parameterone by one (keeping others at nominal) and see how the number of events of theeach sample changes accordingly. From Fig. 5.12, we can clearly see that indeed
the NC1π (bin 71) and NCQE (bin 72) cross-section normalization errors contributemost to the 2Rπ 0 and NCγ de-excitation samples respectively. Also the SK errors
are relatively larger for the NC (and νe CC1π + ) samples.

137

Number of events#toy XP5.3. Systematic parameters

2400
2200
2000
1800
1600
1400
1200

20

15

10

1000
800

5

600
400
200
0
200

220

240

260

0
0

280
300
320
340
Number of predicted events

1

2

3

4

5

6

7

8

9

10

Reconstructed energy (GeV)

Number of events#toy XP(a) Number of predicted events for FHC 1Rµ (b) Number of predicted events in each bin
sample.
for FHC 1Rµ sample.
3000
2500
2000
1500

4
3.5
3
2.5
2
1.5

1000

1
500
0
50

0.5
55

60

65
70
75
Number of predicted events

0
0

1

2

3

4

5

6

7

8

9

10

Reconstructed energy (GeV)(c) Number of predicted events for RHC 1Rµ (d) Number of predicted events in each bin
sample.
for RHC 1Rµ sample.

Figure 5.9: Number of observed events for the 1Rµ samples obtained over 50k throwsof systematic parameters. On the left shows the predicted number of events in all
bins. The dotted lines show the ±1σ region around the mean value. On the rightshows the error envelope in each sample bin. Blue boxes correspond to the ±1σ
region around the mean value, black histograms correspond to the nominal values
for default value of the systematic parameters.

Number of events

2500

2000

1500

#toy XP138#toy XP5.4. Fitting method

20

2500
2000

15
1500
10
1000

1000

5

500

500

0
50

60

70

0
0

80
90
100
Number of predicted events

200

400

600

800

1000

1200

0
5

1400

6

7

Reconstructed pe (MeV/c)

8

9

10

11

12

Number of predicted events

2

Number of events#toy XP

Number of events(a) Number of predicted (b) Number of predicted (c) Number of predicted
events for FHC 1Re sam- events in each pe bin for events for RHC 1Re sample.
FHC 1Re sample.
ple.
9000
8000

1.5

7000
6000

1

5000

3
2.5
2
1.5

4000

1

3000

0.5

2000

0.5

1000

0
0

200

400

600

800

1000

1200

1400

0
0

2

4

6

Reconstructed pe (MeV/c)

8
10
12
14
Number of predicted events

0
0

200

400

600

800

1000

1200

1400

Reconstructed pe (MeV/c)(d) Number of predicted (e) Number of predicted (f) Number of predicted
events in each pe bin for events for FHC νe CC1π + events in each pe bin for
RHC 1Re sample.
sample.
FHC νe CC1π + sample.

Figure 5.10: Number of observed events for the 1Re and νe CC1π + samples obtained
over 50k throws of systematic parameters.

5.4

Fitting method

5.4.1

Likelihood calculation

Suppose in an experiment we observe the numbers of events in each sample bin~ obs := hN b i, we define the joint likelihood function asNobs~ obs , N
~ pred , o, f ) = Lsample (N~ obs , N
~ pred (o, f )) × Lsyst. (f ) × Lexternal (o),L(N(5.17)

where o and f are the oscillation and systematic parameters.
b
In the sample likelihood part Lsample , we quantify the difference between Nobs

Number of events

1000

800

600

#toy XP139#toy XP5.4. Fitting method

2500

20
18

2000

16
14

1500

12
10

1000

8

400

6
500

4

200

2
0
0

20

40

0
0

60
80
100
Number of predicted events

200

400

600

800

1000

1200

0
0

1400

5

10

Reconstructed p 0 (MeV/c)

15
20
25
Number of predicted eventsπ

5

Number of events#toy XP

Number of events(a) Number of predicted (b) Number of predicted (c) Number of predicted
events for FHC 2Rπ 0 sam- events in each bin for FHC events for RHC 2Rπ 0 sample.
2Rπ 0 sample.
ple.
1000

4.5
4

800

3.5
3

600

2.5
2

20
18
16
14
12
10
8

400

1.5

6

1

4

200

0.5
0
0

2
200

400

600

800

1000

1200

0
0

1400

50

Reconstructed p 0 (MeV/c)

100

150
200
250
Number of predicted events

0

5

10

15

20

25

30

Reconstructed gamma energy (MeV)π(d) Number of predicted (e) Number of predicted (f) Number of predicted
events in each bin for RHC events for FHC NCγ de- events in each bin for FHC
2Rπ 0 sample.
excitation sample.
NCγ de-excitation sample.

Figure 5.11: Number of observed events for the NC samples obtained over 50k throwsof systematic parameters.
band Npredin the form of binned Poisson likelihood:− ln Lsample =

X
b

b
b
b
Npred
− Nobs+ Nobs
ln

b
Nobs
,
b
Npred

(5.18)

where b is summed over the event bins from all samples. The oscillation parameters~ pred :and systematic parameters enter in the calculation of N
~ pred (o, f ) =NX

Mb (C, I, Eνtrue )Posc (C, I, Eνtrue ; o)Rb (C, I, Eνtrue ; f ),(5.19)

C,I,Eνtrue

where Mb , Posc and Rb are the non-oscillated event rates, oscillation probabilities
and systematic reweighting factors respectively (see Sec. 5.2.6). Here, the oscillation
parameters o, which are being fit, correspond to θ23 , ∆m231 , θ13 , δCP , θ24 , θ34 andFraction change in number of events

5.4. Fitting method140

FHC 1RµRHC 1R µ

0.2

FHC 1ReRHC 1ReFHC ν e CC1π+
FHC 2Rπ00.15

RHC 2R π0
FHC NC γ de-excitation

0.1

0.05

0
0

20

40

60

80

100
120
Parameter bin

Figure 5.12: Fractional change of number of events when a systematic parameter isincreased by 1σ from nominal value. Parameters are 1-50: beam flux, 51-72: neutrino
interaction and cross-section, 73-132: Super-K detector, 133: SK p-scale parameters.
∆m241 .
The systematic part Lsyst. (f ) is calculated as

Lsyst. = exp(−0.5Xvi Mij vj ),(5.20)

i,j

where vi is the difference of the i-th systematic parameter from its central value and
Mij is the (i, j) element of the inverted covariance matrix.
We use Lexternal to include any external constraint on the oscillation parameters.
For example, in this study, sin2 2θ13 is constrained by reactor measurements, and a5.4. Fitting method

141

Gaussian penalty (x̄ = 0.0857, σ = 0.0046) is used:

Lexternal = exp(−0.5(sin2 2θ13 − x̄)2)σ2

(5.21)In the oscillation analysis, we usually fit for one or two oscillation parameters at
once, and the un-fitted oscillation parameters and systematic parameters are treatedas nuisance parameters. There are several different ways to handle these nuisanceparameters.
The standard way in T2K is to use the method of Bayesian marginalization,i.e. the marginal likelihood is computed by integrating the full likelihood over thenuisance parameters λ:~ obs , N
~ pred , of it ) =Lmarg (NZ

~ obs , N
~ pred , o, f )p(λ(oun−f it , f ))dλ,L(N(5.22)

where of it (oun−f it ) are the fitted (un-fitted) oscillation parameters.
A numerical integration technique is used to estimate the integral, where λ arethrown N = 10000 times according to their prior distributions, and the average L iscomputed:
N
1 X ~
f it
~ pred , o(of it , oun−f it ), fi ).
~~L(Nobs , N
Lmarg (Nobs , Npred , o ) ≈i
N i=1(5.23)On the other hand, in our sterile oscillation analysis, there are three more oscillation parameters in the fit: θ24 , θ34 and ∆m241 . If we use the same method of
marginalization, a much larger number of throws N  10000 is required for the fit
to converge. This takes much more computation time and is not desirable. Therefore
we determine to use the method of minimization instead.

5.4. Fitting method

142

Target coverage 1 parameter
68.3%
1
90%
2.71
95%
3.84
99%
6.63

2 parameters
2.3
4.61
5.99
9.21

Table 5.28: Fixed ∆χ2f values used to build confidence level intervals
We use Minuit2 ROOT package to minimize − ln L with respect to all nuisanceparameters. The Migrad algorithm is used for this minimization, and in case it fails
to converge the Simplex method is used instead.

~ obs , N
~ pred , of it ) = min (− ln L(N~ obs , N
~ pred , o, f )).
− ln Lmin (Noun−f it ,f

5.4.2(5.24)

Confidence level building

A simple way to estimate the confidence level (C.L.) is to use the Wilks’s theorem[167].
We define the ∆χ2 function as:∆χ2 (of it ) = −2 × ln(

Lmin (of it )).
maxof it (Lmin (of it ))(5.25)

The confidence level intervals are the region of parameter space for which the ∆χ2is inferior to a certain fixed value of ∆χ2f (see Table 5.28).

Chapter 6
Sensitivity study
This section describes the sensitivity studies with MC data assuming T2K-SK Run18 proton-on-target (POT), i.e. 1.47341 × 1021 POT in FHC and 0.75573 × 1021 POT
in RHC. For FHC NCγ de-excitation sample, only Run1-4 POT (0.656 × 1021 ) isused. Table 6.1 lists the Run and POT configuration of the eight samples.
We generated a toy MC data set using the set of “true” oscillation parametersin Table 6.2, with all the systematic parameters set at their nominal values. All
statistical fluctuations are suppressed and the numbers of events in each sample bin
are equal to their expected values. This is referred to as the Asimov data set, which isa representative of all possible data sets. The “Asimov Sensitivity” calculated in thisSample
FHC
Run
POT
RHC
Run
POT

1Rµ 1Re 2Rπ 0

νe CC1π +NCγ de-excitation

1-8
1.47341 × 1021

1-8
1.47341 × 1021

1-4
0.656 × 1021

5-7
0.75573 × 1021--

Table 6.1: Run and POT configuration for the eight oscillation samples.

143

144
Parameters
∆m221
∆m232
sin2 θ23
sin2 θ12 (sin2 2θ12 )sin2 θ13 (sin2 2θ13 )δCPEarth matter density
Baseline length
Mass hierarchy∆m241 , sin2 θ24 , sin2 θ34

True values
7.53 × 10−5 eV22.509 × 10−3 eV2
0.528
0.304 (0.846)
0.0219 (0.0857)-1.601
2.6 g/cm3
295 kmNormal0

Table 6.2: Oscillation parameters used to generate ToyMC data set.
Parameters∆m221
sin2 2θ12
sin2 2θ13∆m232
sin2 θ23
δCP

Allowed values
7.53 × 10−5 eV2
0.846
0.0857 ± 0.0046[0.002, 0.003]eV2[0.3, 0.7][−π, π]Table 6.3: Allowed values of the 3-flavor parameters in the oscillation fit.
way can be used to estimate the median significance of many toy MC experiments
which saves a lot of computation time.
Fig. 6.1 shows the predicted event spectra for all samples generated with theassumed POT and oscillation parameters.
In the fitting process, sin2 θ12 = 0.304 and ∆m221 = 7.53 × 10−5 eV2 are held
fixed because T2K lacks sensitivity on them. A penalty term is added to use theconstrains of sin2 2θ13 = 0.0857 ± 0.0046 from [168]. sin2 θ23 , ∆m232 and δCP are
allowed to vary in certain intervals. Table 6.3 summarizes the fitter settings for the
3-flavor parameters.
We perform oscillation fits for both 3-flavor oscillation and 3+1 sterile oscillation

145

θ

180

18

2.5

16

160

14

140

12

120

10

100

8

80

6

60

4

40

2

20

3
2

2.5

1.5

2
1.5

1

0
0

5

10

0
0

15
20
25
30
Reconstructed energy (GeV)

1
0.5

200

(a) FHC 1Rµ

400

600

800
1000 1200 1400
Reconstructed momentum (MeV)

0

0
0

5

(b) FHC 1Re

10

15
20
25
30
Reconstructed energy (GeV)(c) RHC 1Rµ

θ

180

θ

180
160

0.25

160
0.25

140

0.2

0.2

120

100

100

0.15

0.15
80
0.1

0.1

60

40

600

800
1000 1200 1400
Reconstructed momentum (MeV)

6

0.05

2

20
400

10

4

40
0.05

20
200

12

8

80

60

16
14

140

120

0
0

0.5

0

0
0

200

400

600

800
1000 1200 1400
Reconstructed momentum (MeV)

0
0

0

200

600

800

1000

1200

1400

Reconstructed π0 momentum (MeV)(e) FHC νe CC1π +(d) RHC 1Re

400(f) FHC 2Rπ 0

14

3.5

12

3

10

2.5

8

2

6

1.5

4

1

2

0.5
0
0

0

200

400

600

800

1000

1200

5

10

15

20
25
30
Reconstructed γ energy (MeV)

1400

Reconstructed π0 momentum (MeV)(g) RHC 2Rπ 0(h)
FHC
excitation

NCγ

de-

Figure 6.1: Predicted spectra for each sample. Distributions for 1Rµ samples are afunction of Erec (GeV) whereas distributions for 1Re samples and νe CC1π + sample
are a function of pe (MeV) and θ (degree). Distributions for 2Rπ 0 samples are afunction of π 0 momentum (MeV) whereas distributions for NCγ de-excitation sample
are a function of reconstructed γ energy (MeV).

6.1. 3-flavor oscillation fit

146

scenarios.

6.1

3-flavor oscillation fit

A 3-flavor oscillation fit with only the standard CC samples is first done to check the
fitter consistency, then a fit with all eight samples is done to make sure the additional
NC samples do not introduce any strange effect or bias. The 3-flavor oscillation
probability with matter effect is used in the event rate calculation. Details of theevent breakdowns can be found in Table B.1 - B.8 in Appendix B.
Interestingly, the confidence levels with and without the NC samples lay nearly
the same on each other. That means under the current statistics, the NC background of the CC samples is already well constrained by the systematic parameters.
For simplicity, in the following we just show the fit results with all the eight oscillation samples. The best fit point is indicated by the triangle marker. For thetwo-dimensional contours, the normal and inverted hierarchy ∆χ2 are calculatedwith respect to their own χ2min . For the one-dimensional case, a single global χ2min isused.
The sensitivity plots from our analysis are shown in Fig. 6.2-6.4, all together withthe plots from standard analysis[45]. Compared with standard results, our fits havea best-fit point closer to truth, but slightly looser constraints on the fitted oscillation
parameters. This is mostly due to the difference in the likelihood calculation method,i.e. marginalization of L vs. minimization of − ln L over all nuisance parameters.
Yet the two different fitted results are consistent with each other and we conclude
that the introduction of the three NC samples does not have any side-effect on theoscillation analysis.

147∆ m232(eV2)

6.1. 3-flavor oscillation fit

3 ×10

-3

2.9
2.8

NH 90%

NH 68%NH Best Fit

2.7

IH 90%

IH 68%IH Best Fit

2.6
2.5
2.4
2.3
0.4

0.45

0.5

0.55

0.6

0.65

sin2θ23(a) Standard T2K analysis.

(b) Our analysis.

δCP

Figure 6.2: Asimov sensitivity 2D confidence level contours in ∆m232 vs. sin2 θ23 fornormal and inverted hierarchy. The black cross marks the parameter truth values.

32

NH 90%
NH 68%

1

NH Best Fit

0

IH 90%
IH 68%-1

IH Best Fit

-2
-3
16

(a) Standard T2K analysis.

18

20

22

24

26

28

30

32

34

×10-3
36

sin2θ13(b) Our analysis.

Figure 6.3: Asimov sensitivity 2D confidence level contours in δCP vs. sin2 θ13 fornormal and inverted hierarchy. The black cross marks the parameter truth values.

148

∆ χ2

6.2. 3+1 oscillation

18
16
14

NH

12

IH

10
8
6
4
2
0

(a) Standard T2K analysis.

-3

-2

-1

0

1

2

3

δCP

(b) Our analysis.

Figure 6.4: Asimov sensitivity 1D ∆χ2 in δCP for normal and inverted hierarchy.

6.2

3+1 oscillation

In this fit, we use the 3+1 sterile vacuum oscillation probability to calculate the event
rates. With the same set of oscillation parameters in Table 6.2, the 1Re and νe CC1π +event numbers are a few percent different from that in 3-flavor case, while other event
numbers are similar. Table B.9-B.16 in Appendix B shows the corresponding event
breakdowns.

6.2.1

sin2 θ24 -∆m241 plane

We scan the (sin2 θ24 , ∆m241 ) parameter space from 10−3 to 1 in sin2 θ24 and 10−4 eV2
to 103 eV2 in ∆m241 . sin2 θ34 is allowed to float freely from 0 to 1. The best-fit point
is at (sin2 θ24 , ∆m241 ) = (0.0040(0.0023), 0.0048(0.0024)eV2 ) for NH (IH).
Fig. 6.5 shows our expected sensitivity, together with constraints from other experiments. We have a better sensitivity on sin2 θ24 for ∆m241 6 0.003eV2 than otherexisting results, because of T2K’s specific design for ∆m232 -sin2 θ23 precision measurement. The NH outward bump (worse sensitivity) at around ∆m241 = 0.0025eV2 is6.2. 3+1 oscillation

149

due to the degeneracy between m4 and m3 , which does not exist in IH case.
For larger ∆m241 values, we are not that competitive due to statistics and thelack of near detector data. Also, if ∆m241 > 0.3eV2 , sterile oscillation would have
impacts on ND280 at neutrino beam energy peak 0.6 GeV (see Appendix H), andnear detector data needs to be included in the fit (which is not done in this analysis)
to give proper constrains in this region. We therefore only show data fit results only
up to ∆m241 = 0.3eV2 , to avoid any contradiction to BANFF fit assumptions.

∆ m241(eV2)102
10NH 95%
NH 90%

1
10-1

IH 95%
IH 90%

10-2
10-3
10-4 -3
10

10-2

10-1

1

sin2θ24(a) Our analysis.

(b) Constraints from other experiments.

Figure 6.5: Asimov sensitivity 2D confidence level contours in ∆m241 vs. sin2 θ24 fornormal and inverted hierarchy. On the right shows our sensitivity overlaid on limits
to date from other experiments[32–39] (Figure taken from [32]).
Fig. 6.7 illustrates the contribution of different oscillation samples to the fit. At
large ∆m241 , the fit is dominated by the large statistics of 1Rµ events. The νe samples
resolves part of the ∆m241 ∼ ∆m231 degeneracy, and together with the NC samples,
improves the sensitivity at small ∆m241 .
In Appendix I, we show the event spectra at several test points of the 90% limit

6.2. 3+1 oscillation

150∆ m241(eV2)

102
10
1

ν µ samples, 90%
ν µ + ν e samples, 90%
ν µ + ν e + NC samples, 90%

10-1
10-2
10-3
10-4 -3
10

10-2

10-1

1

sin2θ24

Figure 6.7: Asimov sensitivity 2D confidence level contours in ∆m241 vs. sin2 θ24
assuming NH. The exclusion limit is mostly determined by the νµ samples at large∆m241 , and supplemented by νe and NC samples at small ∆m241 .
to illustrate the fitter features.

6.2.2

sin2 θ24 -sin2 θ34 plane

We scan the (sin2 θ24 , sin2 θ34 ) parameter space from 10−3 to 1 in sin2 θ24 and 0 to 1 insin2 θ34 . In the region of ∆m241 > 0.01eV2 , the fit has little dependence on ∆m241 , andwe set a fixed ∆m241 = 0.1eV2 . The best-fit point is (sin2 θ24 , sin2 θ34 ) = (0.001, 0) for
both NH and IH.
Fig. 6.8 shows our expected sensitivity, together with constraints from IceCube[40]and SK[35]. We have looser exclusion limits, especially for sin2 θ34 . Since sin2 θ34
sensitivity comes mostly from NC samples, improvement of NC systematics or development of additional NC samples would be needed if we want to enhance sin2 θ34
sensitivity significantly.

cos2θ24sin2θ34

6.3. Effect of statistics

151

1
0.9

NH 99%

0.8

NH 90%
IH 99%

0.7

IH 90%

0.6
0.5
0.4
0.3
0.2
0.1
0
10-3

10-2

10-1

1

sin2θ24(a) Our analysis.

(b) Constraints from IceCube and SK.

Figure 6.8: Asimov sensitivity 2D confidence level contours in cos2 θ24 sin2 θ34 vs.
sin2 θ24 . On the right shows the 90% and 99% upper limits from IceCube[40] andSK[35].

6.3

Effect of statistics

Our sterile neutrino analysis is still statistically limited at current POT. Fig. 6.9
shows our expected sensitivity with T2K full POT (7.8 × 1021 ) and T2K-II proposed
POT (20×1021 ), where a significant improvement can be seen on the sterile parameterbounds. Also, more oscillation samples are in development by the T2K-SK working
group, e.g. multi-ring νµ /νe samples, and full release of NCγ de-excitation data inboth neutrino and anti-neutrino beam modes. This further increases the statistics
and certainly helps the analysis.

6.4

Effect of systematics

To study the impacts of systematic uncertainties on the analysis, we perform theoscillation fit with different categories of uncertainties switched on and off. Fig. 6.10
shows the effects of systematics due to each of the three error categories. We further

6.4. Effect of systematics

152

Run 8 POT 95%

10

Run 8 POT 90%
Full T2K POT 95%

1

Full T2K POT 90%

cos2θ24sin2θ34∆ m241(eV2)102

T2K-II POT 95%
T2K-II POT 90%

-1

10

-2

10

1

Run 8 POT 99%

0.9

Run 8 POT 90%
Full T2K POT 99%

0.8

Full T2K POT 90%
T2K-II POT 99%

0.7

T2K-II POT 90%

0.6
0.5
0.4

10-3

0.3
0.2

10-4 -3
10

0.1

10-2

10-1

1

sin2θ24(a) sin2 θ24 -∆m241 sensitivity.

0
10-3

10-2

10-1

1

sin2θ24(b) sin2 θ24 -sin2 θ34 sensitivity.

Figure 6.9: Asimov sensitivity 2D confidence level contours of NH assuming current
Run 1-8 POT and future POT.
break down the neutrino interaction and cross-section (Xsec) parameters and the SK
parameters into specific groups and study their relative contributions in Fig. 6.11.
As discussed in Sec. 5.3.4, Xsec and SK parameters have the largest effects on
the newly added NC samples, and affect the sterile sensitivity most. In particular,
the NC1π and NCQE normalization parameters dominate in the Xsec errors. Future
cross-section analysis at ND280 would definitely contribute to systematic reduction.
On the other hand, there will be improvements on the SK errors of NC samples
too. For example, we expect there would be both statistic and systematic enhancements when fiTQun selections of 2Rπ 0 samples are ready for oscillation analysis. For
the NCγ de-excitation sample, the biggest SK error comes from secondary gamma
production induced by knocked-out neutrons. There is an experiment being performed in RCNP, Osaka University [169] to measure the gamma ray emission fromneutron-oxygen interaction, and the results can be used constrain the uncertainties

6.4. Effect of systematics

153

No sys. 90%

10cos2θ24sin2θ34∆ m241(eV2)102

Beam sys. 90%
Xsec sys. 90%

1SK sys. 90%
Full sys. 90%

10-1

1
No sys. 90%
Beam sys. 90%
Xsec sys. 99%
SK sys. 90%
Full sys. 90%

0.9
0.8
0.7
0.6
0.5
0.4

-2

0.3

10

0.2
0.1

10-3

0
10-3

10-2

10-1

-4

10 -3
10

10-2

10-1

1

sin2θ241

sin2θ24(b) sin2 θ24 -sin2 θ34 sensitivity.

(a) sin2 θ24 -∆m241 sensitivity.

Figure 6.10: Asimov sensitivity 2D confidence level contours of NH showing theindividual effect of different sources of systematic uncertainty.
on secondary gamma.

6.4. Effect of systematics

154

10
1

102

∆ m241(eV2)∆ m241(eV2)102
No sys. 90%

10ν µ samples SK sys. 90%

CCQE + 2p2h sys. 90%1ν µ + ν e samples SK sys. 90%-1ν µ + ν e + NC samples
SK sys. 90%

CCQE + 2p2h
+ NC1π/NCQE norm sys. 90%

-1

10

10

10-2

10-2

10-3

10-3

10-4 -3
10

10-2

10-1

1

sin2θ24(a) Xsec parameter contributions.

No sys. 90%

10-4 -3
10

10-2

10-1

1

sin2θ24(b) SK parameter contributions.

Figure 6.11: Asimov sensitivity 2D confidence level contours of NH in sin2 θ24 -∆m241
plane, showing the systematic effects of Xsec and SK parameters.

Chapter 7
Data fit results
In this chapter, we fit for both the 3-flavor and 3 + 1 sterile oscillation models using
T2K Run 1-8 data. Table 7.1 shows the observed event numbers together with the
Asimov predicted ones. Fig. 7.1 shows observed event spectra overlaid on Asimov
predictions. The two sets of numbers and spectra agree quite well, so we expected
the data fit results would be similar to the Asimov fits.
SampleData
Asimov

1RµFHC RHC
240
68
268.4 64.3

1Re
FHC RHC
74
7
73.5
7.9

νe CC1π +
FHC
15
6.9

2Rπ 0
FHC RHC53
949.5 11.3

NCγ de-excitation
FHC
102
107.7

Table 7.1: Event rates of T2K Run 1-8 data and Asimov data set generated usingparameters in Table 6.2.

7.1

Global best fit points

The 3+1 sterile NH (IH) best fit point has a χ2 = 343.8(343.6), while the 3-flavor NH(IH) best fit χ2 = 344.8(348.5). Table 7.2 shows the best fit oscillation parameter

155

7.1. Global best fit points

156

θ

180

18

2.5

16

160

14

140

12

120

10

100

8

80

6

60

4

40

2

20

2

1.5

1

0
0

1

2

3

4

0.5

0
0

5
6
7
8
Reconstructed energy (GeV)

200

400

(a) FHC 1Rµ

600

800
1000 1200 1400
Reconstructed momentum (MeV)

0(b) FHC 1Re
180

θ

8

160

7

0.25
140

6

0.2

120

5

100
0.15

4
80
3

40
0.05

1
0
0

0.1

60

2

20
1

2

3

4

0
0

5
6
7
8
Reconstructed energy (GeV)

200

400

(c) RHC 1Rµ

600

800
1000 1200 1400
Reconstructed momentum (MeV)

0(d) RHC 1Re
25

θ

180
0.25

160

20

140
0.2
120

15

100

0.15

80
0.1

60
40

10

0.05

5

0

0
0

20
0
0

200

400

600

800
1000 1200 1400
Reconstructed momentum (MeV)

200

400

600

800

1000

1200

1400

Reconstructed π0 momentum (MeV)(e) FHC νe CC1π +(f) FHC 2Rπ 0

5

24

4.5

22
20

4

18

3.5

16

3

14

2.5

12

2

10
8

1.5

6

1

4

0.5
0
0

2

200

400

600

800

1000

1200

1400

Reconstructed π0 momentum (MeV)(g) RHC 2Rπ 0

0

5

10

15

20
25
30
Reconstructed γ energy (MeV)(h) FHC NCγ de-excitation

Figure 7.1: Observed event spectra overlaid on the Asimov prediction.

7.2. 3-flavor oscillation
Parameters|∆m232 |(×10−3 eV2 )sin2 θ23
sin2 2θ13
δCP
Mass hierarchy∆m241 (eV2 )sin2 θ24
sin2 θ34

157
3 + 1 sterile
2.43
2.41
0.538
0.542
0.088
0.088
-1.69
-1.45
Normal Inverted
0.012
0.009
0.045
0.047
0.000
0.002

3-flavor
2.46
2.43
0.526
0.531
0.087
0.088
-1.75
-1.46
Normal Inverted-Standard Ptheta
2.46
2.44
0.528
0.533
0.087
0.088
-1.77
-1.44
Normal Inverted-

Table 7.2: Best fit oscillation parameter values for both 3+1 sterile model and 3flavor model. On the right shows the results from standard oscillation analysis [45].
values for all cases. The 3-flavor best fit point are consistent with that of the standard Ptheta analysis. The 3+1 sterile model has a larger sin2 θ23 to allow non-zerosin2 θ24/34 .

7.2

3-flavor oscillation

Fig. 7.2 and 7.3 show the 2D confidence level intervals in ∆m232 vs. sin2 θ23 and δCPvs. sin2 θ13 respectively, and Fig. 7.4 shows the 1D ∆χ2 plots in δCP . They are allconsistent with the standard oscillation analysis results in Ref.[45].

7.3

3+1 sterile oscillation

Fig. 7.5 shows the confidence level contours in the sin2 θ24 vs. ∆m241 plane. Compared to the Asimov fit, the data fit has slightly better exclusion limits on sin2 θ24 for∆m241 < 5 × 10−3 eV2 . For NH, at ∆m241 ∼ 2.5 × 10−3 eV2 , the data fit and Asimov fit
have different shapes (data is inward but Asimov is outward), which seems to be dueto the slight deficit of FHC 1Rµ events in the sub-GeV region (see Fig. 7.1a). Basi-

158∆ m232(eV2)7.3. 3+1 sterile oscillation
3 ×10

-3

2.9
2.8

NH 90%

NH 68%NH Best Fit

2.7

IH 90%

IH 68%IH Best Fit

2.6
2.5
2.4
2.3
0.4

0.45

0.5

0.55

0.6

0.65

sin2θ23(a) Standard T2K result.

(b) Our analysis

Figure 7.2: Data fit 2D confidence level contours in ∆m232 vs. sin2 θ23 for normal andinverted hierarchy.
cally the data fit is consistent with the Asimov fit, with certain statistical fluctuations
that gradually stabilize as ∆m241 > 0.1eV2 .
Fig. 7.6 shows the confidence level contours in the cos2 θ24 sin2 θ34 vs. sin2 θ24
plane, with ∆m241 = 0.1eV2 . The NH and IH contours overlay on each other. Again
the data fit result is consistent with the Asimov ones.

159

δCP

7.3. 3+1 sterile oscillation

3
2

NH 90%
NH 68%

1
NH Best Fit
IH 90%

0

IH 68%-1

IH Best Fit

-2
×10-3
36

-3
16

18

20

22

24

26

28

30

32

34

sin2θ13(a) Standard T2K result.

(b) Our analysis∆ χ2Figure 7.3: Data fit 2D confidence level contours in δCP vs. sin2 θ13 for normal andinverted hierarchy.

30
25

NH

20

IH

15
10
5
0

(a) Standard T2K result.

-3

-2

-1

0

1

2

3

δCP

(b) Our analysis

Figure 7.4: Data fit 1D ∆χ2 plots in δCP for normal and inverted hierarchy.

∆ m241(eV2)7.3. 3+1 sterile oscillation

10-1

160

Data NH 95%
Data NH 90%
Asimov NH 95%
Asimov NH 90%
Data IH 95%
Data IH 90%-2

10

Asimov IH 95%
Asimov IH 90%

10-3

10-4 -3
10

10-2

10-1

1

sin2θ24(a) Our analysis.

(b) Constraints from other experiments.

cos2θ24sin2θ34

Figure 7.5: Data fit 2D confidence level contours in ∆m241 vs. sin2 θ24 for normal
and inverted hierarchy. On the right shows our results overlaid on limit to date fromother experiments[32–39] (Figure taken from [32]).

1
0.9

Data 99%

0.8

Data 90%
Asimov 99%

0.7

Asimov 90%

0.6
0.5
0.4
0.3
0.2
0.1
0
10-3

10-2

10-1

(a) Our analysis.

1

sin2θ24(b) Constraints from IceCube and SK.

Figure 7.6: Data fit 2D confidence level contours in cos2 θ24 sin2 θ34 vs. sin2 θ24 . On
the right shows the 90% and 99% upper limits from IceCube [40] and SK [35].

Chapter 8
Summary and prospect
There are interesting hints from experiments, like LSND and Daya Bay, which suggest
a third mass splitting ∆m2 ∼ 1eV2 in neutrino oscillation. Such results might be
explained by the existence of sterile neutrinos, which are present in many extensionsof the Standard Model. Sterile neutrinos are neutral singlet fermions which do notparticipate in the weak interaction, making direct detection difficult. But if eV-scalesterile neutrinos exist, they can affect the oscillation spectra through mixing withthe three active neutrinos.
In the T2K experiment, the high intensity neutrino beam produces large statisticsof neutrino events which are useful in searching for sterile neutrino. We utilize theoscillation samples in the far detector SK to constrain the sterile mixing parameters
under the 3+1 sterile neutrino model (3 active neutrinos + 1 sterile neutrino). In
particular, we include the NC interaction samples in the oscillation analysis for the
first time to enhance the sterile sensitivity.
The sterile mixing parameters have different effects on the oscillation spectra.
T2K is mostly sensitive to the sterile mixing angles θ24 and θ34 . We modify the T2K

161

162

official fitter to include the sterile oscillation probability and NC oscillation samples.
A joint analysis with both the standard CC oscillation samples and new NC samples
is performed to constrain θ24 and θ34 for a range of ∆m241 .
We include both the CC samples and NC samples in the oscillation analysis. The
standard CC samples, νµ disappearance samples (FHC/RHC 1Rµ) and νe appearance
samples (FHC/RHC 1Re, FHC νe CC1π + ), are sensitive to θ24 . The new NC samples,
FHC/RHC 2Rπ 0 samples (mainly NC1π 0 resonant production) and FHC NCγ deexcitation sample (mainly NCQE events), measure the NC interaction rates of all
three active neutrinos, and give us sensitivity on θ24 and θ34 .
Most systematic parameters inherit from the standard 3-flavor analysis. Two
NC cross-section errors and twelve SK detector errors are added to describe theuncertainties on the newly added NC samples. The SK detector uncertainties arecalculated based on separate studies, with FSI and SI model uncertainties handled
by a reweighting software package.
The fit goodness is described by a joint likelihood function for all samples:~ obs , N
~ pred , o, f ) = Lsample (N~ obs , N
~ pred (o, f )) × Lsyst. (f ) × Lexternal (o),L(Nwhere Lsample is the binned Poisson likelihood between the number of observed events~ obs and the number of predicted events N~ pred calculated with oscillation parametersNo and systematic parameters f . The systematic penalty Lsyst. is evaluated with the
input covariance matrix, and Lexternal is used to include any external constraint on
the oscillation parameters.
An Asimov data set (event rates predicted without any statistical or systematicfluctuations) is generated assuming T2K-SK Run1-8 POT (except NCγ de-excitation

163

sample, only Run1-4), no sterile mixing. The expected sensitivity of the sterile mixing
parameters are evaluated by fitting the Asimov data.
Data fits are performed in two sterile parameter planes: sin2 θ24 vs ∆m241 , andsin2 θ24 vs sin2 θ34 . The data fit results are consistent with the Asimov sensitivity.
Our constraint on sin2 θ24 is better than any existing results for ∆m241 < 0.003eV2 .
However we are not competitive for larger ∆m241 values, or in the sin2 θ34 constraint.
We exclude sin2 θ24 > 0.1 and sin2 θ34 > 0.5 at 90% limit for ∆m241 > 0.1eV2 .
The current analysis is still statistically limited, and future T2K data will definitely improve our results (shown in Sec. 6.3). Additional oscillation samples are indevelopment to provide extra statistics: multi-ring samples which contain higher neutrino energy events and benefit searches at large ∆m241 , and RHC NCγ de-excitation
sample which improves θ34 constraint.
Among the systematic parameters, the newly added NC cross-section errors(NC1π and NCQE) turn out to be the greatest systematic limitations in the fit.
Future cross-section analysis from T2K (or other experiments) might help reducethese errors. The SK errors of NC samples can also be reduced by: the transition
from apfit to fiTQun reconstruction for 2Rπ 0 samples (together with a slight increasein statistics), and the reduction of secondary γ production error with measurementsin RCNP for NCγ de-excitation samples. The extent of improvement can be estimated from the studies in Sec. 6.4.
Another possible improvement strategy is the joint fit analysis with other data
sets. At present, the ND280 (near detector) samples act as the input of BANFF fit
which constrain the neutrino flux and cross-section systematic parameters propagated to the oscillation analysis. The SK detector uncertainties are characterized by theatmospheric neutrino samples, which include some built-in cross-section uncertain-

164

ties. Therefore the cross-section errors are somewhat double-counted and inflate thedetector systematics. A joint analysis that utilizes the T2K-SK oscillation samples,
ND280 samples, and SK atmospheric neutrino samples will both increase the statistics and reduce the systematics. The ND280 samples also provide sterile sensitivity
for ∆m241 up to 100 eV2 . Such analysis framework is now under development by both
T2K and SK members, and will definitely benefit both the standard oscillation fit
and the sterile oscillation fit.
There are some long scale projects related to the T2K. These include the ND280
upgrades[170] and construction of new near detectors in J-PARC (WAGACI/BabyMIND[171], NINJA[172], E61[173–175]) which improves cross-section measurements
and provides new samples for short baseline sterile analysis. In the far detector aspect, we have the SK-Gd upgrade with better neutrino/anti-neutrino discrimination,and the proposed T2HK which uses Hyper-Kamiokande as a new far detector[176](or T2HKK with one more detector in Korea) to gain larger event statistics withsmaller detector errors. The combination of all these efforts contribute to both standard 3-flavor oscillation and sterile oscillation analysis. We look forward to theirrealizations, together with the upcoming short baseline oscillation experiments, thatwould lead to major discoveries in the next five or ten years.

Appendix A
SK event reconstruction
algorithms
The section describes in details the two reconstruction algorithms, apfit and fiTQun,
used in SK.

A.1

apfit

In apfit, the reconstruction process goes through a series of steps:
1. Vertex reconstruction
The event vertex is first approximated by assuming the PMT hits are due to~ is found by minimizing the timea single point source. The point vertex, A,
residual ti for all PMT hits:
ti = t0i −nw ~~|Pi − A|,c

(A.1)

where t0i is raw hit time, and P~i is the position of the i-th PMT. The particle
165A.1. apfit

166

Figure A.1: Illustration of the Hough Transform (Figure taken from [41]).
direction is estimated by summing the PMT hit vectors.
Then, the Cherenkov ring edge is searched by tuning the ring opening angle tomatch the observed charge pattern and improve direction construction.
Finally, the vertex position is refined by considering a line-source of photons
and accounting for scattering effects.
2. Ring counting
After the 1st ring is identified is previous step, the algorithm searches for otherring candidates by applying the method of Hough Transform[177]. Fig. A.1
illustrates the idea. A 42o cone is drawn around each PMT hit, and the intersection of these cones give the ring center.
The ring candidate is then tested by a likelihood function which compares theobserved charge pattern with the N -ring and N +1-ring hypotheses. If theN + 1-ring hypothesis is favored, the process repeats to find the next ring candidate. Otherwise it is stopped and the number of rings is determined to be
N . A maximum of 5 rings can be identified.

A.1. apfit

167

Figure A.2: Illustration of Cherenkov photon emission from a muon. When muon
loss energy in a distance dx, photons are emitted in the region dx · sin θ + r · dθ(Figure taken from [42]).
3. Particle identification
A particle identification (PID) algorithm is applied on each identified Cherenkov ring. The observed charge pattern is compared with the prediction from ashowering event (e-like) and a non-showering event (µ-like) respectively.
The showering events are mostly due to electrons or gammas, where the electromagnetic showers or multiple scattering produce a diffuse charge pattern.
The expected charge distributions are prepared in advance by Monte Carlosimulations, which are characterized by the electron momentum and the PMTangle relative to the electron direction.
For muon rings, muons are assumed to travel in a straight line with no scattering, and the expected charge distribution can be analytically calculated. Also,
when muon loses energy during propagation, the Cherenkov angle slowly decreases as well, which changes the area subtended the Cherenkov photons (seeFig. A.2).

A.1. apfit

168

4. Precise vertex fitting (MS-fit)
The vertex position is refitted by including also the ring pattern information.
The vertex position and particle direction are adjusted to improve the PID
likelihood. The final vertex resolution is around 30 cm for 1-ring events.
5. Momentum reconstruction
For each Cherenkov ring, the particle momentum is reconstructed by summing
the observed PMT charges inside a cone with a half opening angle of 70o . In
case the i-th PMT hit is shared between two or more cones, the charge isseparated based on the expected charge contributions from each ring.
The observed p.e. in the n-th ring are corrected by:



GMC 
α ×
RT OTn =Gdata X
θi,n <70o−50ns<ti <250ns

ricos Θiobs
qi,n
× exp( ) ×Lf (Θi )

X−

θi,n <70o


Si 
,
(A.2)

where
α : normalization factor,
Gdata , GMC : relative PMT gain parameter for data and Monte Carlo simulation,
obs
qi,n: observed charges in the i-th PMT due to the n-th ring,θi,n : opening angle between the n-th ring direction and the i-th PMT direction,
Θi : Angle of photon arriving direction relative to the i-th PMT,
ti : time-of-flight subtracted hit time of the i-th PMT position,
L : light attenuation length in water,A.1. apfit

169

ri : distance from vertex position to the i-th PMT,f (Θ) : correction function for PMT acceptance, andSi : expected charge for the i-th PMT from scattered photons.
RT OT is then converted to momentum using the e/µ conversion table obtained
from MC.
6. Decay electron search
When low energy muons stop in the detector, decay electrons are produced
which are indicator of µ-like events. They are identified by searching for PMT
hit clusters in time (also with some timing and vertex goodness cuts) after
the primary neutrino events. Tagged electrons can be classified into three
categories: primary-event type where the electron is found within the primary
event time window, sub-event type which is outside that time window, andsplit type in which the energy deposition is split between the primary eventand sub-event . The tagging efficiencies are around 96% for µ+ and 80% forµ− in SK-IV.
7. Ring number correction
Some of the mis-identified rings have too low momentum or are overlapped withother rings. The ring number correction process compares the i-th ring withthe j-th ring, where i 6= j. If (I) the i-th ring is of low momentum and overlaps
with j-th ring of larger momentum, or (II) the i-th ring is of low momentum
and the momentum is very small compared with the j-th ring, the i-th ring isremoved.
8. π 0 fittingA.2. fiTQun

170

As mentioned before, single π 0 production in NC interaction constitutes a large
background in the 1-ring e-like samples. This happens when the two gammas
from π 0 decay are considerably asymmetric so one gamma is of too small energy
to be reconstructed, or the two gamma rings overlap and are reconstructed as
one ring only.
A π 0 fitter is applied on the 1-ring e-like events, which assumes the existenceof two overlapping gamma rings. The direction of the more energetic gamma
is fixed as the reconstructed ring direction. The second ring direction and theenergy balance between the two gammas are varied until a best-fit combination
is found. The π 0 rejection is done by a cut on the output likelihood value.

A.2

fiTQunfiTQun is a maximum likelihood fitter that maximizes the following likelihood functionin event reconstruction:

L(x) =

unhit
Y
j

hit
Y
Pj (unhit|x) {1 − Pi (unhit|x)}fq (qi |x)ft (ti |x),

(A.3)

i

where x corresponds to the particle track parameters, P (unhit|x) (1 − P (unhit|x)) is
the PMT unhit (hit) probability, and fq (qi |x) and ft (ti |x) are the probability density
functions of observed charge and time respectively. x can be either a single particle
or multi-particle hypothesis, and the collected charges in each PMT are calculated
analytically based on the MC generated Cherenkov emission profile, light attenuation
factor, PMT acceptance and scattering light contribution.
The reconstruction process is divided into five steps:

A.2. fiTQun

171

1. Vertex pre-fitter
2. Hit clustering
3. Singel-ring fitter
4. π 0 fitter
5. Multi-ring fitter
To begin, a single vertex is first found to avoid the likelihood being stuck at
a local minimum. The vertex pre-fitter uses only the PMT hit time information
to roughly estimate the vertex position, in a way similar to the first step in apfit.
Another option is directly using the apfit vertex.
Next, in hit clustering, a simple algorithm is used to divide the whole event
time window into several “subevents”. Fixing the vertex position as the vertex prefitter’s one, the vertex timing is varied to find peaks in the vertex goodness functions.
For example, in a stopping muon event with decay electron production, the parent
muon production time and the decay electron production time would appear as two
separate peaks. A -180 ns to 800 ns time window is defined around each peak toform a subevent (time windows would be merged into one in case of overlapping),
and the vertex pre-fitter is performed again for each subevent. The decay electron
detection efficiency found in this way is around 88%.
For each subevent, a single-ring fitter is performed to maximize the likelihood
function in Eq. A.3 under the single particle hypothesis. There are three possible
particle hypothesis: e, µ and π + , and the corresponding fitters are applied one by one.
Among them, the most special one is the π + fitter. While the Cherenkov emission
profiles of π + are similar to those of µ because of their similar masses, π + can easilyA.2. fiTQun

172

Figure A.3: Likelihood separation of single-ring electron (left) and muon (right)
events for SK-IV atmospheric neutrino MC. The magenta lines show the cut criteria
for electron-muon separation (Figure taken from [25]).

Figure A.4: Likelihood separation of νµ charged-current quasi-elastic (left) and NCπ +(right) events for T2K neutrino MC. The black lines show the cut criteria for muonπ + separation (Figure taken from [25]).
have hadronic interaction in water which causes scattering or absorption. Thus theπ + fitter fits for two rings: the “upstream-track” ring before hadronic interaction
and the “downstream-track” ring after. The µ fitter result is used as a seed in the
upstream-track reconstruction, and the energy loss Eloss in the upstream-track is fit,assuming constant energy loss per unit length, to give the downstream-track vertex.
The e/µ separation is done by taking the ratio of the log likelihoods ln(Le /Lµ ), andthe µ/π + separation has a two-dimensional cut on ln(Lπ+ /Lµ ) and Eloss . Figs. A.3
and A.4 illustrate the separation powers.

A.2. fiTQun

173

Figure A.5: Likelihood separation of the CC single electron (left) and NC single π 0(right) events in the SK-IV atmospheric neutrino MC. The magenta lines show thecut criteria for electron-π 0 separation (Figure taken from [25]).

Figure A.6: Schematic diagram of ring counting in the multi-ring fitter of fiTQun.
The fifth and sixth rings are assumed to be e-like only. This diagram shows theprocedure assuming the first ring to be e-like, and the same procedure is repeated
for the case of first ring being π + -like (Figure taken from [24]).
The π 0 fitter is a dedicated two-ring fitter to the identify the π 0 → γγ decay. In
each e-like subevent, the e-fit result is acted as the seed for the first gamma ring,and the second gamma ring is searched by minimizing − ln L with respect to ring
directions, momenta and other track parameters. The ratio of the likelihood values
Lπ0 /Le , as well as the reconstructed π 0 invariant mass, are used to reject π 0 frome-like events (see Fig. A.5).
Finally, the multi-ring fitter is applied only on the first subevent, since subsequent subevents usually contain single particle. As shown in Fig. A.6, a search up

A.2. fiTQun

174

to six rings by doing an exhaustive search for all possible ring candidate combinations. Electrons and gammas are fit by e-hypothesis, while muons and π + are fit by
upstream-track π + hypothesis. Starting from the first ring, additional ring is added
assuming the same vertex, and − ln L is minimized over the two-particle track parameters. The log likelihood ratio ln(L2R /L1R ) is evaluate to determine whether the
new ring is true or fake. The process repeats until a fake ring or six true rings arefound. Once the ring counting is done, the rings are re-ordered by visible energy. The
most energetic ring will be merged with lower energy ring(s) if their relative angle isless than 20o . Then all the rings are refitted, in descending order of energy, by thethree particle hypothesis (e, µ, π + ) with other rings fixed. If the ring is identified as
non-showering (µ/π + ), the most energetic ring is always to assumed to be µ while
lower energy rings are π + . This is because νµ CC events usually have a true muonring and very rare a true π + ring is the most energetic in NC events.

Appendix B
Event rate tables
This section lists the event rates of the eight oscillation samples in the sensitivity
study in Section 6.
Event Type νµ → νµCCQE
175.635
CCMEC
35.525CC 1π
27.696
CC coh.
0.288
CC other
5.467
NC 1π
5.523
NC coh.
0.000
NC 1γ
0.000
NC other
2.056
Subtotal
252.191Total

νe → νe
0.004
0.002
0.002
0.000
0.001
0.116
0.000
0.000
0.074
0.199

ν̄µ → ν̄µ ν̄e → ν̄e11.028
0.000
1.409
0.000
2.616
0.000
0.092
0.000
0.406
0.000
0.197
0.011
0.000
0.000
0.000
0.000
0.131
0.009
15.8790.021
268.361νµ → νe
0.022
0.026
0.024
0.000
0.000
0.000
0.000
0.000
0.000
0.072

ν̄µ → ν̄e
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

Total
186.690
36.96130.338
0.381
5.874
5.847
0.000
0.000
2.270

Table B.1: Event rate table for FHC 1Rµ sample in 3-flavor oscillation analysis.

175

176

Event Type νµ → νµ
CCQE
0.161
CCMEC
0.030
CC 1π
0.047
CC coh.
0.000
CC other
0.013
NC 1π
1.870
NC coh.
0.528
NC 1γ
0.942
NC other
0.318
Subtotal
3.910
Total

νe → νe
6.306
1.597
0.932
0.008
0.128
0.042
0.007
0.017
0.017
9.054

ν̄µ → ν̄µ ν̄e → ν̄e
0.006
0.255
0.001
0.043
0.003
0.067
0.000
0.004
0.000
0.009
0.070
0.004
0.048
0.003
0.050
0.002
0.021
0.001
0.200
0.390
73.514

νµ → νe46.282
8.689
4.495
0.034
0.089
0.000
0.000
0.000
0.000
59.589ν̄µ → ν̄e
0.278
0.039
0.047
0.003
0.003
0.000
0.000
0.000
0.000
0.371

Total
53.28910.399
5.592
0.050
0.243
1.987
0.586
1.011
0.357

Table B.2: Event rate table for FHC 1Re sample in 3-flavor oscillation analysis.

Event Type νµ → νµCCQE
15.293
CCMEC
4.470
CC 1π
3.955
CC coh.
0.047
CC other
0.942
NC 1π
0.420
NC coh.
0.000
NC 1γ
0.000
NC other
0.316
Subtotal
25.443Total

νe → νe
0.001
0.000
0.000
0.000
0.000
0.017
0.000
0.000
0.016
0.035

ν̄µ → ν̄µ ν̄e → ν̄e29.874
0.001
2.838
0.000
4.533
0.000
0.176
0.000
0.606
0.000
0.522
0.014
0.002
0.000
0.000
0.000
0.201
0.008
38.7510.023
64.256

νµ → νe
0.001
0.000
0.001
0.000
0.000
0.000
0.000
0.000
0.000
0.002

ν̄µ → ν̄e
0.001
0.000
0.001
0.000
0.000
0.000
0.000
0.000
0.000
0.002

Total
45.171
7.310
8.489
0.223
1.548
0.973
0.002
0.000
0.541

Table B.3: Event rate table for RHC 1Rµ sample in 3-flavor oscillation analysis.

177

Event Type νµ → νµ
CCQE
0.013
CCMEC
0.003
CC 1π
0.008
CC coh.
0.000
CC other
0.004
NC 1π
0.163
NC coh.
0.048
NC 1γ
0.079
NC other
0.057
Subtotal
0.374
Total

νe → νe
0.509
0.144
0.101
0.001
0.021
0.007
0.002
0.004
0.002
0.792

ν̄µ → ν̄µ ν̄e → ν̄e
0.024
0.848
0.002
0.113
0.007
0.144
0.000
0.011
0.001
0.012
0.196
0.005
0.203
0.003
0.175
0.004
0.027
0.001
0.636
1.141
7.920

νµ → νe
1.027
0.233
0.143
0.001
0.010
0.000
0.000
0.000
0.000
1.415

ν̄µ → ν̄e
2.859
0.331
0.334
0.031
0.007
0.000
0.000
0.000
0.000
3.562

Total
5.280
0.825
0.738
0.045
0.055
0.372
0.256
0.263
0.087

Table B.4: Event rate table for RHC 1Re sample in 3-flavor oscillation analysis.

Event Type νµ → νµ
CCQE
0.029
CCMEC
0.010
CC 1π
0.076
CC coh.
0.000
CC other
0.038
NC 1π
0.085
NC coh.
0.000
NC 1γ
0.028
NC other
0.206
Subtotal
0.472
Total

νe → νe
0.024
0.022
0.761
0.018
0.112
0.002
0.000
0.000
0.008
0.948

ν̄µ → ν̄µ ν̄e → ν̄e
0.001
0.002
0.000
0.001
0.003
0.004
0.000
0.000
0.001
0.003
0.005
0.000
0.000
0.000
0.001
0.000
0.013
0.001
0.024
0.011
6.919

νµ → νe
0.212
0.160
4.854
0.111
0.121
0.000
0.000
0.000
0.000
5.458

ν̄µ → ν̄e
0.003
0.001
0.002
0.000
0.002
0.000
0.000
0.000
0.000
0.008

Total
0.270
0.195
5.699
0.129
0.276
0.093
0.000
0.029
0.228

Table B.5: Event rate table for FHC νe CC1π + sample in 3-flavor oscillation analysis.

178

Event Type νµ → νµ
CCQE
0.034
CCMEC
0.009
CC 1π
0.221
CC coh.
0.000
CC other
0.072
NC 1π
32.065
NC coh.
8.639
NC 1γ
0.025
NC other
4.273
Subtotal
45.338Total

νe → νe
0.068
0.015
0.078
0.003
0.021
0.620
0.145
0.001
0.159
1.111

ν̄µ → ν̄µ ν̄e → ν̄e
0.002
0.004
0.000
0.000
0.001
0.005
0.000
0.001
0.000
0.002
1.119
0.062
0.634
0.028
0.001
0.000
0.261
0.021
2.019
0.124
49.473νµ → νe
0.434
0.044
0.373
0.010
0.013
0.000
0.000
0.000
0.000
0.873

ν̄µ → ν̄e
0.002
0.000
0.004
0.001
0.000
0.000
0.000
0.000
0.000
0.007

Total
0.543
0.068
0.683
0.016
0.108
33.866
9.447
0.028
4.714

Table B.6: Event rate table for FHC 2Rπ 0 sample in 3-flavor oscillation analysis.

Event Type νµ → νµ
CCQE
0.002
CCMEC
0.001
CC 1π
0.016
CC coh.
0.000
CC other
0.011
NC 1π
2.168
NC coh.
0.517
NC 1γ
0.003
NC other
0.615
Subtotal
3.333
Total

νe → νe
0.006
0.002
0.008
0.001
0.003
0.088
0.018
0.000
0.028
0.153

ν̄µ → ν̄µ ν̄e → ν̄e
0.005
0.009
0.000
0.001
0.003
0.010
0.000
0.003
0.001
0.002
3.640
0.078
3.308
0.043
0.002
0.000
0.439
0.017
7.398
0.163
11.136

νµ → νe
0.010
0.002
0.010
0.001
0.001
0.000
0.000
0.000
0.000
0.024

ν̄µ → ν̄e
0.028
0.001
0.026
0.007
0.002
0.000
0.000
0.000
0.000
0.064

Total
0.060
0.006
0.073
0.012
0.020
5.973
3.887
0.005
1.100

Table B.7: Event rate table for RHC 2Rπ 0 sample in 3-flavor oscillation analysis.

179

Event Type
CCQE
CCMEC
CC 1π
CC coh.
CC other
NC 1π
NC coh.
NC 1γ
NC other
Subtotal
Beam unrelated
Totalνµ → νµ
2.633
0.000
0.794
0.025
0.108
17.6820.000
0.000
79.115100.357

νe → νe
0.000
0.000
0.002
0.000
0.000
0.361
0.000
0.000
1.114
1.477

ν̄µ → ν̄µ ν̄e → ν̄e
0.059
0.000
0.000
0.000
0.007
0.000
0.002
0.000
0.003
0.000
0.934
0.000
0.000
0.000
0.000
0.000
2.561
0.000
3.565
0.000
2.299
107.698νµ → νe
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

ν̄µ → ν̄e
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

Total
2.692
0.000
0.803
0.026
0.112
18.9770.000
0.000
82.789

Table B.8: Event rate table for FHC NCγ de-excitation sample in 3-flavor oscillation
analysis.

Event Type νµ → νµCCQE
175.739
CCMEC
35.538CC 1π27.702
CC coh.
0.288
CC other
5.467
NC 1π
5.523
NC coh.
0.000
NC 1γ
0.000
NC other
2.056
Subtotal
252.314Total

νe → νe
0.004
0.002
0.002
0.000
0.001
0.116
0.000
0.000
0.074
0.199

ν̄µ → ν̄µ ν̄e → ν̄e11.027
0.000
1.408
0.000
2.616
0.000
0.092
0.000
0.406
0.000
0.197
0.011
0.000
0.000
0.000
0.000
0.131
0.009
15.8770.021
268.478νµ → νe
0.021
0.024
0.022
0.000
0.000
0.000
0.000
0.000
0.000
0.067

ν̄µ → ν̄e
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

Total
186.792
36.97230.342
0.381
5.875
5.847
0.000
0.000
2.270

Table B.9: Event rate table for FHC 1Rµ sample in 3+1 sterile oscillation analysis.

180

Event Type νµ → νµ
CCQE
0.161
CCMEC
0.030
CC 1π
0.047
CC coh.
0.000
CC other
0.013
NC 1π
1.870
NC coh.
0.528
NC 1γ
0.942
NC other
0.318
Subtotal
3.910
Total

νe → νe
6.344
1.605
0.935
0.008
0.128
0.042
0.007
0.017
0.017
9.104

ν̄µ → ν̄µ ν̄e → ν̄e
0.006
0.254
0.001
0.043
0.003
0.067
0.000
0.004
0.000
0.009
0.070
0.004
0.048
0.003
0.050
0.002
0.021
0.001
0.200
0.388
69.154

νµ → νe42.755
8.071
4.205
0.032
0.085
0.000
0.000
0.000
0.000
55.147ν̄µ → ν̄e
0.304
0.042
0.051
0.004
0.003
0.000
0.000
0.000
0.000
0.404

Total
49.825
9.792
5.308
0.048
0.240
1.987
0.586
1.011
0.357

Table B.10: Event rate table for FHC 1Re sample in 3+1 sterile oscillation analysis.

Event Type νµ → νµCCQE
15.296
CCMEC
4.471
CC 1π
3.955
CC coh.
0.047
CC other
0.942
NC 1π
0.420
NC coh.
0.000
NC 1γ
0.000
NC other
0.316
Subtotal
25.446Total

νe → νe
0.001
0.000
0.000
0.000
0.000
0.017
0.000
0.000
0.016
0.035

ν̄µ → ν̄µ ν̄e → ν̄e29.863
0.001
2.837
0.000
4.532
0.000
0.176
0.000
0.606
0.000
0.522
0.014
0.002
0.000
0.000
0.000
0.201
0.008
38.7370.023
64.246

νµ → νe
0.001
0.000
0.001
0.000
0.000
0.000
0.000
0.000
0.000
0.002

ν̄µ → ν̄e
0.001
0.000
0.001
0.000
0.000
0.000
0.000
0.000
0.000
0.002

Total
45.162
7.309
8.489
0.223
1.548
0.973
0.002
0.000
0.541

Table B.11: Event rate table for RHC 1Rµ sample in 3+1 sterile oscillation analysis.

181

Event Type νµ → νµ
CCQE
0.013
CCMEC
0.003
CC 1π
0.008
CC coh.
0.000
CC other
0.004
NC 1π
0.163
NC coh.
0.048
NC 1γ
0.079
NC other
0.057
Subtotal
0.375
Total

νe → νe
0.512
0.144
0.101
0.001
0.021
0.007
0.002
0.004
0.002
0.796

ν̄µ → ν̄µ ν̄e → ν̄e
0.024
0.844
0.002
0.112
0.007
0.144
0.000
0.011
0.001
0.012
0.196
0.005
0.203
0.003
0.175
0.004
0.027
0.001
0.636
1.135
8.216

νµ → νe
0.957
0.219
0.136
0.001
0.010
0.000
0.000
0.000
0.000
1.323

ν̄µ → ν̄e
3.179
0.366
0.365
0.035
0.007
0.000
0.000
0.000
0.000
3.952

Total
5.528
0.847
0.761
0.048
0.055
0.372
0.256
0.263
0.087

Table B.12: Event rate table for RHC 1Re sample in 3+1 sterile oscillation analysis.

Event Type νµ → νµ
CCQE
0.029
CCMEC
0.010
CC 1π
0.076
CC coh.
0.000
CC other
0.038
NC 1π
0.085
NC coh.
0.000
NC 1γ
0.028
NC other
0.206
Subtotal
0.472
Total

νe → νe
0.025
0.022
0.764
0.018
0.112
0.002
0.000
0
0.008
0.952

ν̄µ → ν̄µ ν̄e → ν̄e
0.001
0.002
0.000
0.001
0.003
0.004
0.000
0.000
0.001
0.003
0.005
0.000
0.000
0.000
0.001
0.000
0.013
0.001
0.024
0.011
6.556

νµ → νe
0.196
0.150
4.525
0.103
0.115
0.000
0.000
0.000
0.000
5.089

ν̄µ → ν̄e
0.003
0.001
0.002
0.000
0.002
0.000
0.000
0.000
0.000
0.008

Total
0.255
0.184
5.374
0.121
0.271
0.093
0.000
0.029
0.228

Table B.13: Event rate table for FHC νe CC1π + sample in 3+1 sterile oscillation
analysis.

182

Event Type νµ → νµ
CCQE
0.034
CCMEC
0.009
CC 1π
0.222
CC coh.
0.000
CC other
0.072
NC 1π
32.065
NC coh.
8.639
NC 1γ
0.025
NC other
4.273
Subtotal
45.338Total

νe → νe
0.068
0.015
0.079
0.003
0.021
0.620
0.145
0.001
0.159
1.112

ν̄µ → ν̄µ ν̄e → ν̄e
0.002
0.004
0.000
0.000
0.001
0.005
0.000
0.001
0.000
0.002
1.119
0.062
0.634
0.028
0.001
0.000
0.261
0.021
2.019
0.124
49.414

νµ → νe
0.402
0.041
0.348
0.009
0.012
0.000
0.000
0.000
0.000
0.813

ν̄µ → ν̄e
0.002
0.000
0.004
0.001
0.000
0.000
0.000
0.000
0.000
0.008

Total
0.512
0.065
0.659
0.015
0.108
33.866
9.447
0.028
4.714

Table B.14: Event rate table for FHC 2Rπ 0 sample in 3+1 sterile oscillation analysis.

Event Type νµ → νµ
CCQE
0.002
CCMEC
0.001
CC 1π
0.016
CC coh.
0.000
CC other
0.011
NC 1π
2.168
NC coh.
0.517
NC 1γ
0.003
NC other
0.615
Subtotal
3.333
Total

νe → νe
0.006
0.002
0.008
0.001
0.003
0.088
0.018
0.000
0.028
0.153

ν̄µ → ν̄µ ν̄e → ν̄e
0.005
0.009
0.000
0.001
0.003
0.010
0.000
0.003
0.001
0.002
3.640
0.078
3.308
0.043
0.002
0.000
0.439
0.017
7.398
0.163
11.141νµ → νe
0.009
0.002
0.010
0.000
0.001
0.000
0.000
0.000
0.000
0.023

ν̄µ → ν̄e
0.031
0.001
0.029
0.008
0.002
0.000
0.000
0.000
0.000
0.071

Total
0.062
0.006
0.075
0.012
0.020
5.973
3.887
0.005
1.100

Table B.15: Event rate table for RHC 2Rπ 0 sample in 3+1 sterile oscillation analysis.

183

Event Type
CCQE
CCMEC
CC 1π
CC coh.
CC other
NC 1π
NC coh.
NC 1γ
NC other
Subtotal
Beam unrelated
Totalνµ → νµ
2.633
0.000
0.795
0.025
0.108
17.6820.000
0.000
79.115100.358νe → νe
0.000
0.000
0.002
0.000
0.000
0.361
0.000
0.000
1.114
1.477

ν̄µ → ν̄µ ν̄e → ν̄e
0.059
0.000
0.000
0.000
0.007
0.000
0.002
0.000
0.003
0.000
0.934
0.000
0.000
0.000
0.000
0.000
2.561
0.000
3.565
0.000
2.299
107.699

νµ → νe
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

ν̄µ → ν̄e
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

Total
2.692
0.000
0.804
0.026
0.112
18.9770.000
0.000
82.789

Table B.16: Event rate table for FHC NCγ de-excitation sample in 3+1 sterile
oscillation analysis.

Appendix C
Sterile oscillation probability withmatter effect
In Sec. 6.2, we use the sterile oscillation probability in vacuum in the oscillation analysis. There are concerns that whether the Earth matter would have some resonanceeffect on the oscillation, especially when ∆m241 ≈ ∆m231 . In this section, we check
whether the matter effect has significant impacts on the oscillation probability.
To calculate the sterile oscillation probability with matter effect, we use the Python open-source project NuCraft[178]. One important reason to choose NuCraft isthat it supports arbitrary numbers of neutrino flavors, including sterile neutrinos,
and matter effects with a configurable Earth model. The solution is obtained by numerically solving the Schrödinger equation and can achieve a high enough numericalaccuracy.
In the actual computation, we propagate νµ from the Earth’s surface verticallydown to the neutrino detector at a depth of 295 km, which mimics the T2K-SKbaseline. Two scenarios are considered: constant Earth matter density of 2.6 kg/cm3 ,

184

185
and 0.kg/cm3 (vacuum). We choose three sets of parameter values on the sin2 θ24 ∆m241 plane: {0.3, 10−4 eV2 }, {0.2, 2.51 × 10−3 eV2 } and {0.1, 0.1eV2 }, which roughly
corresponds to the 90% exclusion limit at three different ∆m241 regimes, and examine
the difference in Posc with and without matter effect. As shown in Fig. C.1, matter
effect only induces tiny changes to Pµe and Pµs , and has little effect on our analysis.

(a) {sin2 θ24 , ∆m241 }= {0.3, 10−4 eV2 }(b) {sin2 θ24 , ∆m241 }= {0.2, 2.51 × 10−3 eV2 }(c) {sin2 θ24 , ∆m241 }= {0.1, 0.1eV2 }Figure C.1: Posc with and without matter effect. Unless specified, all oscillation
parameters are the same as in Table 6.2.

Appendix D
Differences between NH and IH at∆m241 ∼ ∆m231
In Sec.6.2, we see that the sin2 θ24 sensitivity curves differ quite a lot when ∆m241 ∼∆m231 . To better understand this result, we plot out Posc with sin2 θ24 = 0.2, ∆m241 =m231 = 2.509 × 10−3 eV2 (other parameters are the same as in Table 6.2). From
Fig. D.1, we can clearly observe that there are sizable differences between NH and
IH, especially for Pµs .
To realize the difference, we can write out Posc explicitly. Neglecting the ∆m221
oscillation terms and assuming θ34 = 0, Posc can be approximated as:∆m231 L
4E
2
∆m
41 L
− sin2 2θ24 (cos2 θ23 + sin2 θ13 sin2 θ23 ) sin2
4E
∆m243 L− cos2 θ13 sin2 θ23 sin2 2θ24 sin2,
4E

Pµµ ≈ 1 − 4 cos2 θ13 sin2 θ23 cos4 θ24 (cos2 θ23 + sin2 θ13 sin2 θ23 ) sin2Pµe ≈ sin2 2θ13 cos2 θ24 sin2 θ23 sin2
186

∆m231
,
4E

(D.1)

(D.2)

187

Figure D.1: Posc with sin2 θ24 = 0.2, ∆m241 = ∆m231 = 2.509 × 10−3 eV2 for NH and
IH.
∆m231 L4E
2
2
2
2
2 ∆m41 L+(cos θ23 + sin θ13 sin θ23 ) sin
4E
2
∆m
43 L
+ cos2 θ13 sin2 θ23 sin2}.
4E

Pµs ≈ sin2 2θ24 {−(cos2 θ23 + sin2 θ13 sin2 θ23 ) cos2 θ13 sin2 θ23 sin2

When ∆m241 ≈ ∆m231 , sin2

(D.3)

∆m243 L∆m2 L
2∆m2 L≈ 0 for NH while sin2 4E43 ≈ sin2 4E31 for
4E

IH. This extra oscillation term with double frequency and amplitude cos2 θ13 sin2 θ23 sin2 2θ24
could modify the oscillation pattern significantly, as illustrated in Fig. D.1. This explains why the exclusion limit of sin2 θ24 differs between NH and IH around this188
∆m241 region.

Appendix E
Effects of θ14, δ14 and δ24 on the fit
In the sterile oscillation fit performed in Sec.6.2, we assume θ14 = δ14 = δ24 = 0.
While this is the usual way similar experiments like MINOS present their results,
it would be interesting to see how the fit results would change if we release thiscondition.
If we allow θ14 , δ14 and δ24 to vary from zero, it would affect the mixing matrix
mostly in the “sterile row” Usi . The full form of Usi without any approximation is:
Us1 = −s12 (−c23 c34 eiδ24 s24 +s23 s34 )+c12 (−c13 c24 c34 eiδ14 s14 −eiδ13 s13 (−c34 eiδ24 s23 s24 −c23 s34 )),(E.1)
Us2 = c12 (−c23 c34 eiδ24 s24 +s23 s34 )+s12 (−c13 c24 c34 eiδ14 s14 −eiδ13 s13 (−c34 eiδ24 s23 s24 −c23 s34 )),(E.2)
Us3 = −c24 c34 e−iδ13 +iδ14 s13 s14 + c13 (−c34 eiδ24 s23 s24 − c23 s34 ),(E.3)

Us4 = c14 c24 c34 .

(E.4)

Consider the third element, Us3 . Given fixed values of the standard 3-flavor

189

190

parameters and θ24 , if we choose the values of θ14 , δ14 and δ24 carefully, we canpossibly make Us3 ≈ 0. Therefore any ∆m231 or ∆m243 oscillation may be made verysmall in the P (νµ → νs ) channel which suppresses the NC disappearance amplitude.
Also, how large the suppression could be would depend on the relative size of s14
and s24 . From Daya Bay Experiment[179], constraints of sin2 θ14 < 0.01 − 0.1 havebeen set over a range of ∆m241 . We may therefore focus on this sin2 θ14 region. In
particular, Daya Bay has released its χ2 map1 . We may define a region of “Daya
Bay constraint” of sin2 θ14 by requiring
χ2 (θ14 , ∆m241 ) − χ2 (3ν) < 2.3,

(E.5)

where χ2 (3ν) is the χ2 of the fit in the 3-flavor model. Fig. E.1 shows this “Dayasin2θ14

Bay constraint” as a function of ∆m241 .

10-1

10-2

10-3

-4

-3

-2

-1

0

1
2
log10(∆ m241 /eV2)

Figure E.1: “Daya Bay constraint” as a function of ∆m241 .
1

https://wiki.bnl.gov/dayabay/index.php?title=Daya_Bay’s_Sterile_Neutrino_
Results_in_2016

191

Below we perform the same sterile oscillation fit as in Sec.6.2, but with free
θ14 , δ14 and δ24 . In particular, we do the fit with 4 scenarios on θ14 : sin2 θ14 = 0,
sin2 θ14 ≤ 0.01, sin2 θ14 ≤ 0.1 and sin2 θ14 ≤“Daya Bay constraint”. Fig. E.2 shows
the results on the sin2 θ24 -∆m241 plane. This mostly changes the constraints of sin2 θ24in the region of ∆m241 . ∆m231 , especially at ∆m241 ≈ ∆m231 . The reason is as
explained: the ∆m231 and ∆m243 oscillation of P (νµ → νs ) is suppressed which removes
the strongest sterile signature in the NC samples.
When ∆m241 ≈ ∆m231 , the situation is more interesting. The non-zero θ14 canreduce the size of Uµ3 = c13 c24 s23 − e−iδ13 +iδ14 −iδ24 s13 s14 s24 , where Uµ4 = c14 e−iδ24 s24
can come in to replace its role as ∆m241 and ∆m231 oscillations are nearly degenerate.
On the other hand, for large ∆m241 , the NC signal is dominated by the fast ∆m241
oscillation where Uµ4 and Us4 are only slighted modified by a factor of c14 , so thefitting results do not change much.
We may present this as a complementary study to the results Sec.6.2.

192∆ m241(eV2)10210sin2θ14=δ14=δ24=0, 90%
sin2θ14=0, free δ14,δ24, 90%1
10-1

sin2θ14≤ 0.01, free δ14,δ24, 90%
sin2θ14≤ 0.1, free δ14,δ24, 90%
sin2θ14≤ "Daya Bay constraint",
free δ14,δ24, 90%

10-2
10-3
10-4 -3
10

10-2

10-1

1

sin θ242

Figure E.2: Asimov sensitivity 2D confidence level contours assuming NH showing
the possible effects of θ14 , δ14 and δ24 .

Appendix F
Setting Exclusion limits of sterile
parameters with the Gaussian CLs
MethodIn Ch. 6, we set the exclusion limits of the oscillation parameters with the fixed ∆χ2method. This method is known to have problems when the fitted parameters areclose to their physical boundaries, e.g. sin2 θ24 = sin2 θ34 = 0, and the confidence
intervals (CI) constructed might not have the correct coverage. We may computethe correct CI using the Feldman-Cousins unified approach [180], but in general it isa very computational-expensive method because of the large number of MC samples
needed for each grid point. Here we would like to try a much simpler way called theGaussian CLs Method which is explained very detailedly in [181].
In the Gaussian CLs Method, we consider pairs of non-nested hypotheses: thestandard 3-flavor oscillation model H0 : β = β0 = {sin2 θi4 = 0}, and the 3+1 sterile
oscillation model H1 : β = β1 = {sin2 θi4 , ∆m241 }, one pair at a time.

193

194

Below we introduce a test statistic, ∆T (β0 , β1 ; x) = ∆T (x), for testing H0 versus
H1 with the data x. We first modify the likelihood functions in Eq.5.17 into:

TH0 (β0 , x; o, f ) = −2 log(L(x, Npred , o(β0 ), f )),(F.1)

TH1 (β1 , x; o, f ) = −2 log(L(x, Npred , o(β1 ), f )).

(F.2)

The subscript H0 (H1 ) means that we use the 3-flavor (sterile) oscillation model inthe event rate calculation.
Then we minimize the modified likelihood functions over nuisance parametersλ(o, f ) to obtain THmin(x) = minλ TH0 and THmin(x) = minλ TH1 . The test statistic ∆T01
is defined as the difference:∆T (x) = THmin(x) − THmin
(x).
1
0

(F.3)

At the first glance, ∆T (x) may be similar to the usual ∆χ2 (x), but in fact they
are different concepts because ∆T (x) involves the best-fit under two different models
H0 (3-flavor) and H1 (3+1 sterile), while ∆χ2 (x) involves the best-fit and the fullparameter space of H1 .
Next we consider the Asimov data set xAsimov
under H0 . Assuming H0 is theH0correct hypothesis, we define∆TH0 = ∆T (xAsimov)
H0= THmin(xAsimov) − THmin(xAsimov
)
H0
H0
1
0
= THmin(xAsimov
).
H0
1

(F.4)

195
Similarly, let xAsimovbe the Asimov data set under H1 , we have
H1
∆TH1 = ∆T (xAsimov)
H1= THmin(xAsimov) − THmin(xAsimov
)
H1
H1
1
0

(F.5)

).
(xAsimov= −THmin
H1
0
Ref. [181] shows that with the following 3 conditions satisfied, ∆T (X) underhypothesis H follows approximately a Gaussian distribution with mean ∆TH andqstandard deviation 2 |∆TH |, where H could be either H0 or H1 :CD1. The parameter space of the nuisance parameters are continuous and themodel likelihood function is a smooth function within the parameter space.
CD2. The data size is large enough (the total number of events is greater than100).
CD3. The predictions by the best model under the null hypothesis H0 and thealternative hypothesis H1 are relatively small.
We define the CLs value as:

CLs (x) =1 − p1,
1 − p0(F.6)

where 1 − p1 (1 − p0 ) is the probability that a single experiment will yield a ∆T (X)value larger than ∆T (x) when H1 (H0 ) is true. With this definition, a CLs value
close to zero would favor H0 against H1 , while it needs to be much larger than one
to have H1 favored against H0 . The exclusion space is typically defined as the set ofparameter value with CLs value smaller than α = 0.05 [182].
With CD1,2,3 satisfied, we may use ∆TH to calculate the probability:

196


1 + Erf√

8|∆TH0 |1 − p0 ≈2
1 + Erf

1 − p1 ≈where Erf(s) = √22∆TH0 −∆T (x)∆TH1 −∆T (x)

√

,

(F.7)

,

(F.8)



8|∆TH1 |2

R s −t2
e dt is the Gaussian error function.
0

Therefore the CLs value can be approximated by
1 + Erf
CLs ≈√8|∆TH1 |
1 + Erf∆TH1 −∆T (x)∆TH0 −∆T (x)

√


.

(F.9)

8|∆TH0 |Fig. F.1 shows the exclusion limits of our sterile oscillation analysis drawn by
the CLs method, using the same configuration in Sec. 6.2. The CLs 95% exclusion
limit is a bit tighter than the fixed ∆χ2 90% limit, which is as expected because theCLs test statistic ∆T is designed for distinguishing the differences between the newphysics hypotheses (sterile oscillation with sin2 θi4 > 0) and the Standard Model (3flavor oscillation with sin2 θi4 = 0). So when the true sin2 θi4 is indeed zero, a strong
limits can be set. On the other hand, if the true sin2 θi4 is deviated from zero a lot(the difference between H0 and H1 is large), CD3 is violated and the CLs method isless useful.
As discussed in [181], although the CLs method is a restricted method that aims
at setting upper limits only, its simplicity makes it a popular approach to present
searches for new physics beyond the Standard Model. Also, the CLs method can
set comparable exclusion limits as the Feldman-Cousins approach on the parameter
space when the new physics hypotheses are close to the Standard Model, which meets

197

10
1
-1

10

NH Fixed ∆ χ2 95%
NH Fixed ∆ χ2 90%
NH CLs 95%
IH Fixed ∆ χ2 95%
IH Fixed ∆ χ2 90%
IH CLs 95%

cos2θ24sin2θ34∆ m241(eV2)

102

10-2

1
0.9

Fixed ∆ χ2 99%

0.8

Fixed ∆ χ2 90%
CLs 95%

0.7
0.6
0.5
0.4

-3

10

0.3
0.2

10-4 -3
10

0.1

10-2

10-1

1

sin2θ24(a) ∆m241 -sin2 θ24 exclusion limits.

0
10-3

10-2

10-1

1

sin2θ24(b) cos2 θ24 sin2 θ34 -sin2 θ24 exclusion limits.

Figure F.1: Exclusion limits of the sterile parameters drawn by CLs and fixed ∆χ2methods.
our needs in the sterile neutrino search. It might be good to present the CLs limits
together with the fixed ∆χ2 limits as a cross-check of results.

Appendix G
Fitting non-Asimov data sets
The Asimov data set fit in Ch. 6 estimates the median sensitivity to the oscillation
parameters. Here we fit for non-Asimov data sets to test the stability of the fitter.
First, for 2500 non-Asimov data sets, we determined the global best-point point
for each data set by minimizing the likelihood function in Eq. 5.17 with respect toall oscillation parameters. Fig. G.1 shows the best fit χ2 distributions for both 3+1
sterile and 3-flavor oscillation models. For the sterile model, NH and IH has similar
best fit χ2 , while 3-flavor model has slightly larger ones, especially for IH. Fig. G.2
shows the best fit standard parameter values. While there are certain shifts of the
distributions, it mostly agrees with the true values. Fig. G.3 shows the best fit
sin2 θ24 -sin2 θ24 distribution. Basically the best fit points concentrate around the 0,0
region, with a small probability spreading out.
Next, we fit for several non-Asimov data sets to see the variations of the sensitivity
curves. Table G.1 lists the event rates of four non-Asimov data sets. Among them,set 1 is the most special: it has event rates smaller than nominal for every sample,
while other data sets have more “random” fluctuations in the event rates. Figs. G.4

198

199

600

180
160

3+1 sterile NH

140

3-flavor NH

120

3+1 sterile IH

100

3-flavor IH

NH

500

IH

400
300

80
200

60
40

100

20
0

250

300

350

400

450

0
-10

-5

0

Best fit χ2

5

10Best fit χ215

3-flavor

20

25- Best fit χ230

sterileFigure G.1: Best fit χ2 distributions of both 3+1 sterile and 3-flavor oscillation
models for 2500 non-Asimov data sets.
Data set
Asimov
non-Asimov 1
non-Asimov 2
non-Asimov 3
non-Asimov 4

1Rµ
FHC RHC
268.4 64.3
190
40
275
69
321
74
254
54

1Re
FHC RHC
73.5
7.9
67
8
66
10
56
6
82
9

νe CC1π +
FHC
6.9
4
7
6
6

2Rπ 0
FHC RHC
49.5 11.3
37
10
46
8
58
17
37
6

NCγ de-excitation
FHC
107.7
78
99
85
119

Table G.1: Event rates of four non-Asimov data sets.
and G.5 show the exclusion limits of the non-Asimov data sets, drawn by the fixed∆χ2 or CLs method. As expected, non-Asimov set 1 prefers non-zero θ24 /θ34 , whileothers have limits quite similar to the Asimov sensitivity. Within small statistical orsystematical fluctuations, the fitter is stable enough to give reasonable results.

200

350

250

300

3+1 sterile NH

3+1 sterile NH

200

3-flavor NH

250

3-flavor NH

3+1 sterile IH

200

3+1 sterile IH

150

3-flavor IH

150

3-flavor IH

100

100
50

50
0
0.4

0.45

0.5

0.55

0.6

0.65

0.7

0
0.00220.00230.00240.00250.0026

Best fit sin2θ23

0.00270.0028

Best fit ∆ m232
600

180
160

3+1 sterile NH

140

3-flavor NH

120

3+1 sterile IH

100

3-flavor IH

500

3+1 sterile NH
3-flavor NH

400

3+1 sterile IH
3-flavor IH

300

80
200

60
40

100

20
0
0.083 0.084 0.085 0.086 0.087 0.088 0.089 0.09 0.091

0

-3

-2

-1

0

1

Best fit sin22θ13

2

3Best fit δCP

Best fit sin2θ34

Figure G.2: Best fit standard oscillation parameters for 2500 non-Asimov data sets.

1

200

0.9

180

0.8

160

0.7

140

0.6

120

0.5

100

0.4

80

0.3

60

0.2

40

0.1

20

0
0

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

0Best fit sin θ24
2

Figure G.3: Best fit sin2 θ24 -sin2 θ34 distribution for 2500 non-Asimov data sets.

201

10

102

∆ m241(eV2)∆ m241(eV2)102
Asimov 90%
Non-Asimov-1 90%

1

Non-Asimov-2 90%

10

Non-Asimov-1 CL s 95%

1

Non-Asimov-4 90%

10-1

10-2

10-2

10-3

10-3

10-4 -3
10

10-2

Non-Asimov-2 CL s 95%
Non-Asimov-3 CL s 95%

Non-Asimov-3 90%

10-1

Asimov CL s 95%

10-1

1

sin θ242

10-4 -3
10(a) Fixed ∆χ2 method.

Non-Asimov-4 CL s 95%

10-2

10-1

1

sin2θ24(b) CLs method.

1
0.9

cos2θ24sin2θ34

cos2θ24sin2θ34

Figure G.4: Exclusion limits in the ∆m241 -sin2 θ24 plane from the Asimov and nonAsimov data sets, assuming NH.

Asimov 90%
non-Asimov-1 90%

0.8

non-Asimov-2 90%
non-Asimov-3 90%

0.7

non-Asimov-4 90%

0.6

1

0.8
0.7
0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0
10-3

10-2

10-1

(a) Fixed ∆χ2 method.

1

sin2θ24

Asimov CLs 95%
Non-Asimov-1 CL s 95%
Non-Asimov-2 CL s 95%
Non-Asimov-3 CL s 95%
Non-Asimov-4 CL s 95%

0.9

0
10-3

10-2

10-1

1

sin2θ24(b) CLs method.

Figure G.5: Exclusion limits in the cos2 θ24 sin2 θ34 -sin2 θ24 plane from the Asimov
and non-Asimov data sets, assuming NH.

Appendix H
Valid ∆m241 parameter space
In the sensitivity study and data fit of the sterile analysis, we limit our fits in regions
below ∆m241 = 0.3eV2 . This cut is determined by the value of ∆m241 beyond which
the near detectors (ND) start to see sterile oscillation.
Our oscillation measurement strategy is to use the ND data to predict the nonoscillated event rates at far detector, and fit for the oscillation parameters fromthe event rate spectral distortions. Since both the non-oscillated event rates andthe systematic parameters are determined by the ND BANFF fit, we cannot simply
remove the BANFF constraints and claim the result is valid. In an even deeper level,
many of the pre-BANFF fit neutrino cross-section parameters come from oscillation
experiments with near detectors at a distance from sources, so they can also beaffected by sterile oscillation. To solve these issues, T2K is developing a “moregeneral” cross-section model which is potential free of these artefacts. But before
this and the related short baseline sterile analysis are done, we have decided not toprovide any result in the problematic region.
To see how we determine the cut value, Fig. H.1 shows, e.g. P (νµ → νµ ) at ND

202

203
at 0.6 GeV (the neutrino beam peak), as a function of ∆m241 with sin2 θ24 = 0.1.
We can see sub-percent changes beyond 0.3 eV2 . We also performed a test study byusing PSK (νµ → νµ )/PN D (νµ → νµ ) and PSK (νe → νe )/PN D (νe → νe ) in event rate
calculation. The changes to sensitivity are shown in Fig. H.2, which is insignificant
below 0.3eV2 . This shows that our cut is well-justified.
1.00

PHΝ Μ®Ν ΜL

0.98
0.96
0.94
0.92
0.90
0.01

0.1

1

Dm241HeV2L

10

100

Figure H.1: P (νµ → νµ ) at ND at 0.6 GeV, as a function of ∆m241 with sin2 θ24 = 0.1.

204

∆ m241(eV2)10210Nominal 95%
Nominal 90%

1

Alternative 95%
Alternative 90%

10-1
10-2
10-3
10-4 -3
10

10-2

10-1

1

sin2θ24

Figure H.2: Asimov sensitivity 2D confidence level contours in ∆m241 vs. sin2 θ24 ,assuming NH. The blue lines show the nominal sterile analysis, and the red lines show
a test study of using PSK (νµ → νµ )/PN D (νµ → νµ ) and PSK (νe → νe )/PN D (νe → νe )in event rate calculation. This illustrates that out sterile fit is valid below ∆m241 =0.3eV2 where the ND sees little sterile oscillation.

Appendix I
Event spectra at sterile exclusion
limits
To illustrate how the sterile exclusion limits are set by the oscillation samples, weshow the event spectra at three specific points on the sin2 θ24 -∆m241 plane: Point #1{0.3, 10−4 eV2 }, Point #2 {0.2, 2.51 × 10−3 eV2 } and Point #3 {0.1, 0.1eV2 }, whichroughly corresponds to the 90% exclusion limit at three different ∆m241 regimes. We
can clearly see the sterile effects on the samples: excess at small ∆m241 and deficit atlarge ∆m241 for 1Rµ samples, and varying amount of deficit for νe and NC samples.

205

206

25

25

20

20

15

15

10

10

5

5

0
0

1

2

3

4

5
6
7
8
Reconstructed energy (GeV)

0
0

200

400

600

800

1000

1200

1400

Reconstructed pe momentum (MeV)(a) FHC 1Rµ(b) FHC 1Re

8

5

7

4.5
4

6

3.5
5

3

4

2.5
2

3

1.5
2

1
1
0
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0.5
1

2

3

4

5
6
7
8
Reconstructed energy (GeV)

0
0

200

400

600

800

1000

1200

1400

Reconstructed pe momentum (MeV)(c) RHC 1Rµ(d) RHC 1Re

10

25

9
8

20

7
6

15

5
4

10

3
2

5

1
0
0

200

400

600

800

1000

1200

1400

0
0

200

400

600

800

1000

1200

1400

Reconstructed π0 momentum (MeV)

Reconstructed pe momentum (MeV)(e) FHC νe CC1π +(f) FHC 2Rπ 0

5

18

4.5

16

4

14

3.5

12

3

10

2.5
8

2

6

1.5

4

1

2

0.5
0
0

200

400

600

800

1000

1200

1400

Reconstructed π0 momentum (MeV)(g) RHC 2Rπ 0

0

5

10

15

20
25
30
Reconstructed γ energy (MeV)(h) FHC NCγ de-excitation

Figure I.1: Asimov event spectra (black lines) together event spectra at Points #1(blue), #2 (red) and #3 (green), assuming NH, overlaid on data points (black points).
The νe spectra are projected on the electron momentum pe .

Appendix J
List of abbreviations and symbols
1Re

Charged-current 1-ring e-like samples

1RµCharged-current 1-ring µ-like samples

2Rπ 0

Neutral-current 2-ring π 0 -like samples

2p2h(2 particle-2 hole) Multiple nucleon interaction

νMSM

Neutrino Minimal Standard Model

apfit

Reconstruction algorithm in SK

fiTQunReconstruction algorithm in SK

p.e.

photo-electrons

BeRPA

Bernstein parameterization of random phase approximation

BANFF fit

Neutrino flux and cross-section fit by the Beam And ND280
Flux extrapolation task ForceBONSAI

Low energy event reconstruction tool in SK

207

208

BSM

Beyond Standard Model

CC interaction

Charged-current interaction

CC1π

Charged-current interactions associated with single-pion resonant production

CC coherent

Charged-current coherent pion production

CCQE

Charged-current quasi-elastic interaction

CP

Charge-parity

FC(Event) Fully contained in ID

FHC

Forward Horn Current (neutrino beam mode)

FSI

Final state interactions

FV

Fiducial volume of detector

ID

Inner detector of SK

IH

Neutrino inverted mass hierarchy (m3 < m1 < m2 )INGRIDInteractive Neutrino GRID (near detector of T2K)

KS testKolmogorov-Smirnov test

LE events

Low energy events

NCγ de-excitation

Neutral current sample with gamma de-excitation events

NC interaction

Neutral-current interaction

NC1π

Neutral-current interactions associated with single-pion resonant production

209

NC1γ

Neutral-current interactions with single gamma production

NC coherent

Neutral-current coherent pion production

NCQE

Neutral-current quasi-elastic interaction

ND280

Main near detector in T2K

NEUT

Neutrino interaction generator used in SK and T2KNH

Neutrino normal mass hierarchy (m1 < m2 < m3 )ODOuter detector of SK

PMNS matrix

Pontecorvo-Maki-Nakagawa-Sakata neutrino mixing matrix

PMT

Photomultiplier tube

POT

Proton-on-target

RFG

Relativistic Fermi gas model

RHC

Reverse Horn Current (antineutrino beam mode)

RPA

Random phase approximation

SF

Spectral function model

SI

Secondary interactions

SK

Super-Kamiokande

SKDETSIM

Detector simulator of Super-Kamiokande

Xsec parameters

Neutrino interaction and cross-section interaction parameters

θc

Cherenkov opening angle

210

θij

Neutrino mixing angle

Uαi

Neutrino mixing matrix element∆m2ij

Neutrino mass splittings m2i − m2j

δCPCP-violating phasenhitacNumber of PMT hits in the largest outer detector hit cluster

wall

Closest distance between vertex and detector wall

towallParticle track length in ID

EνrecReconstructed neutrino energy

Evis

Sum of reconstructed particle momentum of each Cherenkov
ring

ovaQ

Goodness of fit in BONSAI

N30

Maximum number of PMT hits in 30 ns time window

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